Ok, before we start I need to clear something up. The names 'Rat' and 'mouse' are only common names and as such are used to describe many different rodents from unrelated family groups. Therefore they are NOT scientific classifications. In fact the words 'rat and mouse' are just common names to group together rodents which look alike to the casual eye.
For example, the name 'Rat' is used to describe medium-sized rodents with long thin tails, and to confuse matters more, there are many species of rodent that are called rats:
Norway rats - otherwise known as the Brown rat
African pouched rats
Naked mole rats
Pack rats - not to be confused with Rat packs which have no relation to actual rats whether they can sing or not!
Polynesian rats, and many others besides.
The name 'Mouse' is used to describe tiny, sparrow-sized rodents with long thin tails. As with rats, there are many species of rodents called mice which may or may not be closely related to each other:
Dormice - a popular dish with the Romans when roasted in clay
So, which rats and mice are you talking about? Generally, people are referring to the domestic or pest rats and mice, which means Norway rats (Rattus norvegicus), black rats (Rattus rattus), and house mice (Mus musculus).
So, just what is the difference between a rat and a mouse?
Be that as it may, the descendants of that common ancestor diverged into different species, among which are Norway rats and house mice.
Norway rats and house mice now have many genetic, reproductive, developmental, morphological and anatomical differences. The list below is not exhaustive, but for those with a casual interest it should get you started:
Norway rats have 22 chromosome pairs, house mice have 20. Norway rats have 2.75 million base pairs while mice have 2.6 million (humans have 2.9). About 90% of rat genes have counterparts in the mouse and human genomes.
In general, Norway rats develop more slowly than house mice. For example, Norway rat gestation is slightly longer (21-24 days) than house mouse gestation (19-20 days). Norway rats lactate for about 3 weeks, house mice for 2 weeks. Both species are born naked and blind, but Norway rats open their eyes at 6 days, they are fully furred at 15 days. House mice open their eyes at 3 days, have fur at 10 days (etc.).
Norway rats have 6 pairs of nipples, house mice have 5 pairs.
Correlated with this larger size, Norway rat body parts are larger than those of the house mouse -- rats have larger ears, feet etc. The heads of Norway rats are heavy, blunt and chunky, house mouse heads are small and sharply triangular with pointed muzzles. Note, however, that Norway rats have smaller ears relative to their heads than house mice.