Sony Vegas:
Pros - Easy to use, Fast to edit with (do edits quickly), Sony stabilize works pretty well to stabilize shaky footage (not as good as warp stabilizer in after effects, but it works)
Cons - basic effects, terrible color correction, masking is overly complicated
Adobe After Effects:
Pros: Can do pretty much ANYTHING with just one program, for larger projects (100GB+) the render preview makes it so that the edit doesn't "lag" and you can time transitions properly, easily utilize other adobe programs (export frames to photoshop, use premier and export a portion to after effects, then re import etc)
Cons: hard to learn how to use, time consuming
I don't really use premier that much so i cant really comment on it.
Each program does different things well. At the end of the day if i want a perfect final product and i have unlimited amounts of time i will use After Effects, if I want something quick - Vegas it is