Difference between mice and rats
Mice (Mus) and rats (Rattus) are definitely separate species, by whichever definition of species you choose to use. They are genetically distinct, and both genera are members of the clade Muridae, along with Acomys ("spiny mice") and Gerbillus ("gerbils").

Moreover, what we call colloquially "mice" and "rats" encompass a great deal of variation within rodents. The genus Mus is one of several subgenera with ~15 species. 

A recent phylogenetic study of rodents by Blanga-Kanfi et al. (2009), revealed that Mus and Rattus are sister groups, meaning that each is the other's closest relative. The image below from the paper above shows the complete tree. Muridae is about 2/3 of the way down in blue.