Perhaps the most important distinction between the two is that this site is not a discussion forum. From Szabolcs's email to the group announcing this site's launch:
This website is not a traditional discussion forum but a collectively
edited questions and answers site: ideally each question thread will
eventually become a little piece of knowledge that is of use not only
to the original asker but also to everyone who finds it using a search
So questions that are mostly rants (e.g. https://mathematica.stackexchange.com/q/1917/5, which already has one vote to delete) or soliciting strong opinions/bullet-list answers (e.g., What is the strength of Mathematica compared to MATLAB, which is now deleted so you need 2k+ in beta to view), which would've gotten an "Amen, bother!" response or led to opinionated arguments on MathGroup are explicitly off-topic on this site.
That is not to say that the same question cannot be asked constructively (e.g.: Implementing efficient multiple undo) and we don't salt the earth if you use the word MATLAB or its concepts either (e.g. How can I improve the speed of eigenvalue decompositions for large matrices?, which ultimately turned out to be an issue with
Timing or Why don't * and ^ work as I expected on matrices?). We have also been a little relaxed in allowing some level of discourse rather than an extremely strict Q&A with one definite answer, but these questions are exceedingly rare (e.g. Placement of Condition /; expressions).
In addition, there's always the option of using the Mathematica chatroom if you want to engage in discussions (pretty much about anything, not necessarily Mathematica), ask/talk about stuff that is off-topic for the main site etc. People have also known to help others with small issues without having to ask a question formally on the main site. Your students can also pop in to ask if certain kinds of questions would fly on the main site. Sometimes it might be a dupe and regular users might fish it out for them. The rules in the chatroom are very relaxed and the place is a lot more informal, but civility is certainly a must :)
Cross-posting between MathGroup and this site is fine. There is a higher likelihood of finding niche domain experts (not necessarily Mathematica experts) on MathGroup because of its age, but it would be courteous to users of both if you explicitly mentioned that this has been cross-posted, with links to each, so that people can check to see if an answer has been posted before writing one of their own. In the same spirit, it would be courteous if you let people know if you have gotten a satisfactory solution on the other group. If the solution was on MathGroup, then I'd suggest recommending to the answerer to also post it here. Barring that, the OP can answer it themselves and attribute it to them and link to that thread.