First of all, I like the Banshee project. However, I've just been continually underwhelmed by its features compared to Rhythmbox. They both read and write tracks to my IPhone, which is very nice. They both monitor my music folder for new music. Great. They both integrate with the Ubuntu One music store, which I like.
There are some differences, though. Rhythmbox ships with a plugin on Ubuntu that allows DAAP sharing. With Banshee, I have to run Tangerine separately. Also, Rhythmbox seems to be able to list the songs in an album in the correct order more often than Banshee. I don't know why, but even albums that have proper tagging sometimes play in the wrong order in Banshee. Banshee leaks memory. I left it open over the weekend (default configuration, no extra plugins enabled) and when I got back it was using 1.3 GB of memory. Not only that, but I had to kill the application as it wouldn't exit cleanly. Rhythmbox, on the other hand, has a very low memory profile, rarely going above 80 MB even when left running for days.
I like Banshee, but version 1.7.6 that ships with Ubuntu 10.10 still doesn't quite seem ready to replace Rhythmbox. Maybe if I get bored enough, I'll D/L the source and poke around and see if I can't help fix these shortcomings. The idea of tracking down memory leaks in a C# app doesn't seem like a lot of fun, though. Mono is a fine platform, I just haven't done much with it in awhile.
UPDATE: Well, I've reviewed the situation, and there are a couple votes in the opposite direction. First of all Rhythmbox fails to transfer songs successfully to my IPhone, creating invalid or unplayable entries. Conversely, Banshee does this correctly. Also, Banshee has Amazon MP3 store integration that works nicely. You buy the tracks and they download into your collection instantly. That's nice. Also, the other day I noticed that Rhythmbox's memory usage managed to creep up to nearly 400 MB, so it obviously isn't perfect in this respect. For now, I think I'll try using Banshee, I'll just have to remember to restart it periodically to keep memory usage sane.

