I have no practical experience here to draw on, so I'll just tell you what my UG thesis advisor told me when I was choosing schools. A good fit with a professor is the most important part of grad school. The actual school itself doesn't matter much, as long as the professor is a good supervisor and you're interested in their work. That being said, I'm in Canada, and the rankings/prestige/etc of schools seems to play less of a role in future career/academic prospects here than in the U.S. So if you're from the U.S., maybe somebody else has different advice for you.
Also, as a final note, you mentioned the stipend was better at School A. If money is really going to be an issue for you, then I would take that into consideration as well. I get very anxious about money, so it is an important factor in my decisions. But, if it's not going to be a big issue, then I wouldn't prioritize it as much.