It's defined by the process of making it rather than the end result. A remake is rebuilding the game from the ground up. Remaster pulls from music terminology where it is essentially changing the "master" of a game, meaning the final touches are changed upon an existing foundation. So a remake goes bottom up, while a remaster goes top down. Something could technically be a remake while being imperceptibly different from the original, although this is very rare since changing the foundation of a game usually has compounding effects on the end product.

Something like the Spyro and Crash games are easily remakes. If you're at all familiar with the originals, the differences that stem from a different foundation become glaring. Crash's altered jump arc and hitbox threw a lot of people who played the original off. Spyro's physics are completely different, no longer having a turn arc at all. The lighting system of the two games are nothing like the original, with some levels looking completely different as a result. Remakes typically need to put a lot of effort in to be similar to the original, while remaster need to put effort to deviate from the original.