Hello Frederic!
To "pick up" in this context (there are quite a few other meanings), means to physically go to a place to collect something that has been left somewhere, or that is needed.
To "fetch" in this context means to go and pick something (or someone) up and bring that thing or person back.
So the difference is that the first one is just to collect and the second is to collect and bring back.
I hope that helps!