Bloody William
First time out of the vault

I'm not here to pick on anyone who's adamantly anti-Fallout in this thread, I just want to make the case that Fallout 3 is, in fact, both a valid sequel and a worthy successor to the first two Fallout games.
Fallout 3 is a drastic departure in gameplay mechanics. I won't deny that. It's also a buggy game with stability issues. I won't deny that either. However, in terms of tone, design, and general gameplay it is a Fallout game through and through.
What do you think of when you think of Fallout? I think of the SPECIAL system, the Vault Experiment, the feeling of being a lone chump fresh out of the Vault/village and trying to make sense of why everything you know just fucked up and why there's so much resting on your shoulders. I think of ghouls and supermutants and the Brotherhood and scavengers and raiders and dozens of tiny, hilarious little in-game jokes where any trope or story is fair game. I think of walking through ruins and scavenging whatever you can find. I think of choices, and seeing a half dozen possible ways to solve a problem from diplomacy to gunshots. I think of a wasteland full of things both vital to your quest and completely irrelevant, but still entertaining and sometimes useful. I think of an open-ended, paced plot where you can screw around and do whatever you want (granted, with the time limit of the first half of Fallout), but then you go after the main quest and see a major plot development that both changes and greatly increases the responsibility you have, moving from saving your family to saving the entire Wasteland. And all of those aspects, Fallout 3 carries in spades.
Story-wise, this is a Fallout game. Yes, it's different from Fallout and Fallout 2, but Fallout 2 was different from Fallout, and the differences between the three Fallouts are far, far less than the differences between any of those games and FOT/BoS. It's on the east coast now, and the difference between the Capital Wasteland and the west coast both justify and explain the differences in certain aspects of the game. You're just out of your little paradise, looking for your father in a quest that turns into something much bigger. In that time, several towns and hot spots offer you plenty of other stuff to do, not railroading you in any way to stick with the main plot. You can negotiate peace between costumed lunatics, hunt zombies, recover ancient pre-war artifacts, help people with researching whatever crazy crap they're researching, encourage junkies' drug addictions, investigate other Vaults and the crazy things the Vault Experiment had planned for them, and more. The BoS is different in attitude, but that's because (and this is explained clearly) it's a liberal splinter group that headed east and decided to stay and actually help the survivors, not just hoard tech. If you want the classic BoS, they're still out west, and can be seen in the Outcasts, who are far more traditional BoS soldiers than the ones at the Citadel. Supermutants are different, and they're supposed to be. This is explained in-game that, yes, supermutants were made in the same was as the west coast supermutants, but there are reasons that they're different. It makes more sense than the same supermutants as Fallout and Fallout 2, and results in a far better, more Fallout-y game than if they were completely missing.
The humor and storytelling is still there, and frankly in far better condition than in Bethsoft's other work. This stuff is leaps and bounds above most of the things in the Elder Scrolls games. The main quest is about as solid as the previous main quests (not amazingly so, but good enough to play through), the NPCs, settings and different quests bring a great humor that ranges from the chuckle worthy to the ridiculously dark, and anywhere from Rivet City to Megaton to the Citadel, you're looking at the same sort of environ seen in Fallout and Fallout 2, settlements and hazards that developed by people looking for ways to survive in any way they can after the war. There's just as much out there to find, and some of it is as bleak as you would find in any Fallout game.
[spoiler:7a15df2feb]Tenpenny Tower, anyone? Do "the right thing" and negotiate a peace between the residents and the ghouls, and the ghouls eventually slaughter everyone. Tranqulity Lane is a mad German genius's (the same genius who created the G.E.C.K.) attempt to stay alive and entertained after the way, through virtual reality and the deranged torture of the few survivors still alive from pre-war. Andale is home to cannibal murderer "nuclear families." Minefield and other ruins have abandoned houses with skeletons in some of the most stark, dark tableaus you'll see in a video game.[/spoiler:7a15df2feb]
Obviously both the first and second games were built around the SPECIAL system, which was expressed through a turn-based hex grid for combat and movement. Sure, I remember and even enjoyed the turn-based combat. When you got skills up to a decent point it was very satisfying (assuming the combat area wasn't populated with mobs and NPCs so each turn took like five minutes). Yes, Bethsoft changed that, but between the implementation of the SPECIAL system and the clever use of VATS to make combat far more deliberate and less real-time than the Elder Scrolls games, they've come up with something entirely new that works surprisingly well. No, it's not turn-based, but it still has me considering position of myself and enemies, number of AP I have, and the range/capacity of my gun far more deliberately than if it was simply an FPS. You can run into a fight and gun everyone down, but it'll take longer, you'll waste ten times as many bullets, and you'll take a lot more damage. Or you can use some Fallout turn-based-ish strategy and pick your shots, take down some targets and then protect yourself/hide/use cover while your AP recharges to finish up the fight. It's not the same thing, but it works well and it doesn't interfere with the flow of the game. If anything, it's faster while keeping much of the same elements, down to the "Vrrrrrt" sound of VATS, the same sound as the little combat button in Fallout and Fallout 2.
I don't know what Fallout is to you. Maybe it's the turn-based combat, in which case there are plenty of other, indie games that are more worthy of being called Fallout sequels. But if you're looking for a really good RPG that keeps the Fallout spirit and humor, that looks and feels like Fallout, that sticks reasonably close to canon (better than FOT and BoS at the very least), and that's full of great things you'll only see in the Wasteland, then Fallout 3 is a true and worthy sequel in the series.
As for the gameplay differences, look at the difference between Final Fantasy XII and the other, semi-turn-based (real-time but discrete and menu-based) Final Fantasies (not counting 11, which is an MMO). Look at the difference between the Super Mario Bros. games and Mario 64. Look at the difference between the early Castlevania games and Symphony of the Night (regarded by many, including myself, to be the best Castlevania game and one of the best games of all time, period). You can turn gameplay mechanics on their ears and still have a really, really good sequel that satisfyingly continues the series.
Fallout 3 is a drastic departure in gameplay mechanics. I won't deny that. It's also a buggy game with stability issues. I won't deny that either. However, in terms of tone, design, and general gameplay it is a Fallout game through and through.
What do you think of when you think of Fallout? I think of the SPECIAL system, the Vault Experiment, the feeling of being a lone chump fresh out of the Vault/village and trying to make sense of why everything you know just fucked up and why there's so much resting on your shoulders. I think of ghouls and supermutants and the Brotherhood and scavengers and raiders and dozens of tiny, hilarious little in-game jokes where any trope or story is fair game. I think of walking through ruins and scavenging whatever you can find. I think of choices, and seeing a half dozen possible ways to solve a problem from diplomacy to gunshots. I think of a wasteland full of things both vital to your quest and completely irrelevant, but still entertaining and sometimes useful. I think of an open-ended, paced plot where you can screw around and do whatever you want (granted, with the time limit of the first half of Fallout), but then you go after the main quest and see a major plot development that both changes and greatly increases the responsibility you have, moving from saving your family to saving the entire Wasteland. And all of those aspects, Fallout 3 carries in spades.
Story-wise, this is a Fallout game. Yes, it's different from Fallout and Fallout 2, but Fallout 2 was different from Fallout, and the differences between the three Fallouts are far, far less than the differences between any of those games and FOT/BoS. It's on the east coast now, and the difference between the Capital Wasteland and the west coast both justify and explain the differences in certain aspects of the game. You're just out of your little paradise, looking for your father in a quest that turns into something much bigger. In that time, several towns and hot spots offer you plenty of other stuff to do, not railroading you in any way to stick with the main plot. You can negotiate peace between costumed lunatics, hunt zombies, recover ancient pre-war artifacts, help people with researching whatever crazy crap they're researching, encourage junkies' drug addictions, investigate other Vaults and the crazy things the Vault Experiment had planned for them, and more. The BoS is different in attitude, but that's because (and this is explained clearly) it's a liberal splinter group that headed east and decided to stay and actually help the survivors, not just hoard tech. If you want the classic BoS, they're still out west, and can be seen in the Outcasts, who are far more traditional BoS soldiers than the ones at the Citadel. Supermutants are different, and they're supposed to be. This is explained in-game that, yes, supermutants were made in the same was as the west coast supermutants, but there are reasons that they're different. It makes more sense than the same supermutants as Fallout and Fallout 2, and results in a far better, more Fallout-y game than if they were completely missing.
The humor and storytelling is still there, and frankly in far better condition than in Bethsoft's other work. This stuff is leaps and bounds above most of the things in the Elder Scrolls games. The main quest is about as solid as the previous main quests (not amazingly so, but good enough to play through), the NPCs, settings and different quests bring a great humor that ranges from the chuckle worthy to the ridiculously dark, and anywhere from Rivet City to Megaton to the Citadel, you're looking at the same sort of environ seen in Fallout and Fallout 2, settlements and hazards that developed by people looking for ways to survive in any way they can after the war. There's just as much out there to find, and some of it is as bleak as you would find in any Fallout game.
[spoiler:7a15df2feb]Tenpenny Tower, anyone? Do "the right thing" and negotiate a peace between the residents and the ghouls, and the ghouls eventually slaughter everyone. Tranqulity Lane is a mad German genius's (the same genius who created the G.E.C.K.) attempt to stay alive and entertained after the way, through virtual reality and the deranged torture of the few survivors still alive from pre-war. Andale is home to cannibal murderer "nuclear families." Minefield and other ruins have abandoned houses with skeletons in some of the most stark, dark tableaus you'll see in a video game.[/spoiler:7a15df2feb]
Obviously both the first and second games were built around the SPECIAL system, which was expressed through a turn-based hex grid for combat and movement. Sure, I remember and even enjoyed the turn-based combat. When you got skills up to a decent point it was very satisfying (assuming the combat area wasn't populated with mobs and NPCs so each turn took like five minutes). Yes, Bethsoft changed that, but between the implementation of the SPECIAL system and the clever use of VATS to make combat far more deliberate and less real-time than the Elder Scrolls games, they've come up with something entirely new that works surprisingly well. No, it's not turn-based, but it still has me considering position of myself and enemies, number of AP I have, and the range/capacity of my gun far more deliberately than if it was simply an FPS. You can run into a fight and gun everyone down, but it'll take longer, you'll waste ten times as many bullets, and you'll take a lot more damage. Or you can use some Fallout turn-based-ish strategy and pick your shots, take down some targets and then protect yourself/hide/use cover while your AP recharges to finish up the fight. It's not the same thing, but it works well and it doesn't interfere with the flow of the game. If anything, it's faster while keeping much of the same elements, down to the "Vrrrrrt" sound of VATS, the same sound as the little combat button in Fallout and Fallout 2.
I don't know what Fallout is to you. Maybe it's the turn-based combat, in which case there are plenty of other, indie games that are more worthy of being called Fallout sequels. But if you're looking for a really good RPG that keeps the Fallout spirit and humor, that looks and feels like Fallout, that sticks reasonably close to canon (better than FOT and BoS at the very least), and that's full of great things you'll only see in the Wasteland, then Fallout 3 is a true and worthy sequel in the series.
As for the gameplay differences, look at the difference between Final Fantasy XII and the other, semi-turn-based (real-time but discrete and menu-based) Final Fantasies (not counting 11, which is an MMO). Look at the difference between the Super Mario Bros. games and Mario 64. Look at the difference between the early Castlevania games and Symphony of the Night (regarded by many, including myself, to be the best Castlevania game and one of the best games of all time, period). You can turn gameplay mechanics on their ears and still have a really, really good sequel that satisfyingly continues the series.