is the Rattus genus, to which all "true" rats belong, whereas Mice belong to the , or Mus genus. These genera are both subgroups to the sub-family , or Murinae.
Now, these are the scientific names, and luckily, in contrast to English, these scientific names are based on Japanese words, so they do make sense for common people. If you really need to distinguish between the two it depends on your situation. If you are just talking in general to common people, for rat you could either say , or simply describe it , and for mouse you could say , or describe it .
In a more scientific situation, or if you think that the other parties of the conversation know about rats and mice, then you should use the appropriate name for that particular species instead (if you have to distinguish between them), as the above mentioned and are just typical examples of rats/mice, but not necessarily the particular species you saw/bought/whatever.