Species: Norway rats and house mice belong to different species. A species is a group of related individuals or populations which are potentially capable of interbreeding and producing fertile offspring. So Norway rats and house mice belong to different species and cannot interbreed. Humans and orangutangs, chipmunks and red squirrels, bottlenosed dolphins and killer whales, all belong to different species.
Similarities and Differences: Norway rats and house mice are related, however. They descend from a common ancestor that lived millions of years ago -- how long ago is currently under debate, with estimates ranging from 8 to 41 million years ago. That estimate will probably become more precise over time. 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:
Genetic differences: Norway rats have 22 chromosome pairs, house mice have 20 (see Levan 1991). Norway rats have 2.75 billion base pairs while mice have 2.6 billion (humans have 2.9). About 90% of rat genes have counterparts in the mouse and human genomes (Rat Genome Sequencing Consortium 2004). See Burt et al. 1999, Grutzner et al. 1999, and Watanabe et al. 1999 for more.
Growth differences: 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.).
Anatomical differences: Norway rats have 6 pairs of nipples, house mice have 5 pairs.
Morphological differences: Norway rats are larger, heavier and longer than house mice (Norway rat: 350-650 grams, 9-11 inch bodies and 7-9 inch tails; house mice: 30-90 grams, 3-4 inch bodies and 3-4 inch tails). 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.