Mafia: general discussions.

How do the different types of attacks differ? Is it just to create a bunch of different types of defenses and add even more randomness to the game?

Well, not randomness, but rather to encourage more thoughtful selection of skills. Each role has its own characteristics and defenses against certain types of attacks.
In 108, for example, there were characters with immunity to physical attacks, but they could be killed with mental attacks, or vice versa.
Here, it's something similar, but more complex diverse.