Having had neither but having read everything about them that I have come across, I would point out that the 770 seems to be essentially the 710 with a new name, and while some people say the 710 has given them satisfactory service, many people report defects and disappointment and the consensus is that it is inferior to competitor's products offered for significantly lower prices. The Mossberg has not been around as long, but it is basically another version of their ATR100 which most users report as being not all a disappointment like the 710 was.
Given the choice between those two, I'd say "Why can't I have a Stevens 200?"