The brain bots specifically use brains as the cpu, but the process of extracting the brain wipes the personality from it. So it's not good for much else.
The Think Tank seems to have gotten around this, albeit, only partially, since the process of being a brain in a jar seems to unhinge them just a tad. They have a vague sense of self and of what they were like as humans (it's holding up a handful of erect penises!). Memories are incomplete and easy to mess with by messing with their onboard clocks and maps.
The body specifically plays an instrumental role in NV canon in the formation of personality and a person is just fine even if you just replace the brain with an electrode or some other mechanical substitute. That's really the humdinger. The brain isn't everything in NV. Fallout Science!
Obviously a lot of that Old World Blues stuff is pretty silly though, so take it with a grain of salt.
House specifically kept his body intact because the brain-in-the-jar wasn't a perfect solution for immortality.
I think Mister Handy's are supposed to be much less sophisticated, but they honestly never give you that impression in NV and robots like ED-E have full on AI's and vague glimmerings of personality.