(Pure) Mathematician- Stresses more on the theoretical and abstract aspects of mathematics. Deals more with theorems and their proofs than computations. (Usually, when you say "mathematician", you usually mean this type)
Applied Mathematician- Stresses more on the applications of mathematics to some other fields. Deals more with computations and algorithms than theorems and proofs.
Note that these are very simplified distinctions between those two professions. As for grad school, I think pure math would be good if you're thinking of going to grad school in math, and applied math would be good if you're thinking of going to physics (but again, that's probably not the case all the time).