The essential difference is not whether the time is added before or after the move. Rather, it is whether the full amount is added each move (increment), or whether the full amount or time actually used, whichever is less, is added (delay).
With a 5-second delay, if you use only 2 seconds to make your move, you don’t get to keep the remaining 3 seconds. With a 5-second increment, you do. So with increment, your remaining time can actually increase with each move.
A 5-second delay fits in well with fast time controls, such as game/60. An increment (usually 30 seconds) is more suited for longer controls. For example, a tournament formerly played at game/120 may nowadays be played instead at game/90 with a 30-second increment.