As sort of a general guideline, a short bat or crop is designed to be used behind your leg, after putting your reins in one hand, and as discipline or a wake up call for a horse that's not listening to your leg aid. You can also use a short bat or crop to give a sharp tap on the shoulder, sort of a "Hey, pay attention!" but horses, particularly school horses, get dull to that pretty quickly.
A dressage whip is longer so it can be used behind the leg without you taking a hand from the reins or breaking the contact. A dressage whip is also more flexible in its use; you can just tickle the horse behind your leg with the lash or be quite severe with it. The length of the dressage whip makes it difficult to control when galloping, jumping or going cross country or hacking out.
I tend to use a short bat for discipline and the dressage whip more for refining response to leg aid, since you don't drop the contact when using it.
So, if I'm working in the ring on a fairly forward horse that needs a tune up = dressage whip.
Lazy horse that's behind my leg = could use either; would probably start with the bat or crop
Jumping, hacking out, cross country = short bat
Something to carry in case of emergency that I can keep quiet and still when I don't need it = short bat
Hope that helps.