In fact, the main themes of La Bohème become side stories in Rent. Most of the other connections have sort of an in-joke feel. The single riff that Roger is stuck on happens to be Musetta's Waltz from - La Bohème.
The major concept in La Bohème is that love is more powerful than money or survival. Musetta (Maureen) keeps coming back to Marcello (Mark) despite constant arguments and her love of money from rich boyfriends. Mimi cannot survive the squalid conditions under which Rodolfo (Roger) lives, but comes back to him and dies, rather than staying healthy and warm with a rich viscount.
In Rent, Marcello is split into Mark and Joanne. Mark's story becomes one of disconnection, of avoiding life by hiding from it behind a camera lens - and beginning to come out of it. It's actually a bit hard to believe that Maureen ever hung around with him.
The Joanne/Maureen story is also quite different. Musetta is La Bohème herself - she's her own woman, does what she wants, loves who she wants - and is still a good woman. This is a revolutionary idea for the period. Marcello is constantly arguing with her mostly because she, well, does who she wants. In contrast, Joanne and Maureen simply don't get along because they're incompatible personalities - but love each other insanely anyway.
The Collins/Angel affair is a new addition. The two characters exist in the La Bohème, but they certainly don't sleep together. Collins is Colline, the philosopher (Big change there), and Angel is the musician Schaunard, somewhat of a side character. The addition, of course, has several functions. Rent now has (1) one of each possible sexual connection, (2) a relationship that works, and (3) can directly confront death and loss.
And then there's the Roger/Mimi story. Rodolfo and Mimi are central to La Bohème, and there are strong similarities in the story, but the differences are profound. First of all, Puccini's Mimi is largely a good girl, an innocent, who is unfortunately drawn to Rodolfo. He comes to realize that her health will not survive his living conditions, and so simulates jealousy in order to drive her away. He succeeds for a while, and she hooks up with a rich viscount, but cannot stay away - and comes back to Rodolfo to die.
By contrast, in Rent, Mimi is far from an innocent - and is dying anyway. Roger, again, acts jealous in order to cover up his motivation - but he's hiding it from himself. He's afraid to see her die. He runs away to mythical Santa Fe, but is drawn back to look for her - and she is found, well, not-quite-dying. In fact, in this case Roger is more a source of life than the inadvertant cause of death.
It is quite funny to follow the two stories as they develop. The happenings and settings are almost identical, but the stories are in many ways completely different. The new story brings in: Mortality (Through AIDS), homelessness, homosexuality, drug use, and much deeper characters. Certainly the viewing of one is no particular excuse not to see the other.
Larson also brings one more thing to Rent that, as a nerd, I just have to comment on. The man obviously knows nothing about computers or virtual reality. The show was written towards the end of the dot-com explosion, so I can see a certain amount of his disaffectation, but I find it odd how little he knows about the subject.
What the hell would a "virtual-reality" studio be, anyway? Also, how in the world do you type "Angel" on an ATM keyboard?
Ah well. That's my rant for the moment. There's an incredible amount more that could be said, but real life beckons.