What are the differences between them and in which way should we use each?
Stack exchange is a question and answers site, not a forum. It is meant to be used for focused questions that require a concrete answer (more on this on the [help] section, or on the [tour]). It is not a place for discussions, opinions or showcase your work.
Blenderartists.org is a forum with many specialized sub-forums, there you can ask questions as well, and discuss your opinions and voice out your grievances. Note that threads on that site can go on an on, so finding concrete information it is not as simple and straightforward as on this site. Blenderartists.org has more traffic and participants. But the signal-to-noise ratio can be harder to navigate.
So if you just want a simple answer, use the stackechange site. For mixed topics and random conversations use the other one.