We're talking academic music, artsy types here, right? Orchestra type (the band instruments are orchestral too?). As opposed to any form of popular music from the past several decades (though jazz is an exception). Just checking, because what I'm going to say applies to that, and not to rock, pop, metal, country, or any of those popular music types. I'm gonna label it "academic music" (because "classical music" doesn't quite fit).
I went to an arts magnet school for high school. Students auditioned to get into one of five arts. I was in music. In one of my elective courses, which was sociology and psychology, my teacher shared with us something that a previous student had put together. It was a generalized observation of how each type of student dressed at the school - the visual artists looked like works of art themselves (very colourful), the drama students wore loose clothing (and not a lot of it) as if ready to change wardrobe on a moment's notice, etc. As for the musicians, we were described as "wearing armour, very conservative clothing, a form of protection from the world (as if the instruments themselves are a form of armour)". And I think that was a very accurate description.
It wasn't just about the clothing, but I feel like these observations really described how we interacted with people as well. Academic musicians are generally more reserved, or at least, even if they are a more extroverted/outspoken type of person, there is still something there that they hold back a bit, for protection. They keep a little distance between themselves and others.
Now, as for what would have two groups of musicians up against one another, right off the top of my head some form of rivalry would be appropriate. I mean, they could be competing for the practice room (or stage) time, or to be the "headline act" at concerts that their school puts together (each art at my high school had some form of evening event twice a year, and for music, it was a concert with each class - band, orchestra, choir, string ensemble - and each extracurricular music group too). There could be an outside competition as well (like, we had a city-wide music competition that we competed in each year... and yeah, while we were in different categories - like SATB choir, female choir, trios, duets, etc - we still got feedback from the judges... so even if your two music groups both win their category in a competition like this, they could go based on judge comments, or even scores, if you introduce scores into it, to see which group did better each year).
So in this case, it doesn't matter if they're rich or not - you can have the competition be something where wealth doesn't matter, only talent and performance do. Ah, and as for being rich, rich people think in a different way than middle class and poor people do. They view money and time in a different way, they value it differently, and the energy they put around it is different too. It's actually a lot more positive than you might expect.
And that could work in with the rivalry - time is more valuable than money, it's not something you can get more of (unlike money), and once your time is done, it's gone and you can't get it back... so, if their rivalry involves the time they get to spend in the practice room(s), their attitude toward that precious and valuable time can certainly play a part. If one group reserves the prime time slot for practice and the other has to settle with a more inconvenient time slot, that could certainly create tension. Especially if one group often gets the jump on the other for the best practice times each week (if reservations are made on a weekly basis). Or, if one group has that time assigned to them (for the year), and the other complains about it and tries to work it out with administration to get them in there (and perhaps they do get in there on a trade-off basis, each week alternating or something, this can also cause tension between the groups, especially the group that feels their time was "stolen" from them). Etc.