They are not the same. Bush babies are very bushy. That's why they are named that. Bush babies have big tailes.
EDIT: They are different forms and Genuses entirely. Additionally Bushbabies do not...
The first mouse lemur to be described by science was the grey mouse lemur in 1777. However, Buffon mentioned the "Rat de Madagascar" in 1776, which was a reference to a mouse lemur.