It's been brought up countless times, and not only was it a complete failure as a weapon, but the Fatman is still far too different to be the same thing, the Davy Crocket was a tripod mounted weapon with a far higher yield warhead.
The Fatman is supposed to be catapult launched, which means the nuclear payload is propelled by the weapon itself and not its own propulsion system, thus for the weapon to explode the nuclear reaction has to occur when it simply strikes the target from such a weak physical propulsion.
Not only does it make any sense in real life, but it also doesn't make much sense in the Fallout world, not necessarily because it's a nuclear weapon being used in a post-nuclear environment, but rather because the majority of weaponry in Fallout is rare, scrounged and repaired from what was found, the fact that something like the Fatman is capable of operation and is found easily throughout the wastes is absurd.
This has been mentioned multiple times before, and people have presented far more compelling arguments than my simple assessment.