A thriller is more interested in suspense, putting you "on the edge of your seat" or in plain terms, making you nervous for the fate of the characters. A horror movie wants to make you fear for your personal safety, to make you afraid that what happened to the characters may happen to you even after you've left the theater.
As a good horror movie often contains a great deal of suspense and as a good thriller can be pretty scary in spots it's more down to the intentions of the film maker and the subject matter of the film. Hitchcock is regarded as a suspense director but his films Psycho and The Birds are more correctly classified as horror films because they deal with horrific subject matter and their intention is to terrify, not make you nervous about the fate of the on-screen characters. Hitchcock's film North By Northwest has a terrifically suspenseful ending but it is not particularly horrific in that it is not specifically designed to scare you into thinking you may fall off Washington's nose on your next visit to Mount Rushmore.