The wild rats and fancy rats are not of different species and hence the difference isn’t so stark that you can make out easily. However, when you tame a wild animal for generations, there are bound to be differences in their appearance and their behavior, which can be identified by a keen eye.
The fancy rats tend to be a bit pudgy with visible rolls and contours on the body, thanks to relatively comfortable lifestyle and readily available food. Wild rats on the other hand are streamlined, as they have to survive in the wild. They tend to look big only because of puffing of the fur, but one can’t see rolls on their body. A wild rat is generally of variegated brown color or solid black, however the fancy rats are found in whites, beiges, tans, browns, peaches, greys and blacks.
Behavior wise, fancy rats are more social and comfortable around humans, however wild rats will always tend to run away from you. Also, fancy rats can be generally found in relaxed position or motionless in their cage or at home, while the wild rats are always on the move and ready to respond to any threat.