What's the difference between the two ways to create games against the computer in Starcraft 2?

Solution 1:

"Play versus A.I." is a single player mode, you can NOT get achievements, it IS available during offline play.

"Custom game" is a multiple player mode, where you can play against A.I's or humans, or a mix of both. during this mode you CAN get achievements, but you MUST be connected to battle net the whole time.

Solution 2:

One important aspect of a custom game is the ability to choose options before the game starts such as map, speed and in particular race. If you want to practice your TvP on Cloud Kingdom vs a hardest AI at fastest speed you can select exactly those conditions. Player vs AI is random map and random opponent.