Beside the obvious visual differences.... Mice eat seeds and a few insects. Rats will eat anything alive that they can overcome, and only if hungry will they turn to seeds etc. This is ofcourse, in the wild.
Another interesting thing about rats is that they are one of the few animals (along with some humans) that they will eat a balanced diet if it's available......for example if someone throws a burger away they will eat part of the bun, lettuce and other veg, and the meat whereas most animals dog for example would just eat one type of food to the exculsion of all others if given the choice.
Having kept both mice and rats as pets in the past.... I've come to the conclusion that rats are alot more intelligent, clean and friendly.
My rat Patsy comes to her name, only poos in one corner of her hutch and loes nothing better then a belly rub!
This is of course domestic rodents... but im sure the clean and intelligent part still counts in the wild.