This is a tricky one. I'd seriously consider applying for both positions - until you have an offer in hand, you don't have a job. Things may change such that A or B is suddenly unable to offer you their position for whatever reason, and if you haven't applied for both you may find yourself out in the cold. Also, there may be details to each position (pay, duties, opportunities for advancement) that you don't know about, and you won't know about unless you apply and interview. Once you have all of the actual details from interviewing, and the details of whatever offers you actually receive, then choose the one that seems best for you. But don't rule out either one before you actually have a concrete offer - after all, things have been up in the air all summer and could change again!
As for what to tell people if you get offered both positions, I'd stress that it was a very difficult decision, you feel very lucky to have had two excellent choices, but you could only pick one. Notice that that's not really an answer to why you chose B over A - most people don't actually need that answer from you and should be deflected rather than answered. For what to tell A, I'd still stress that it was a very difficult decision, you would have greatly enjoyed either position, but you ended up going with B because of the slightly higher compensation (as LeveeWoman said, more money is hard to argue with when choosing a job - it's one of the main purposes of working).
And congrats on having such a difficult decision to make!