I"ve been to both quite a few times and I would unhesitatingly pick Carmine's over R on R every time. I like the atmosphere at Carmine's better and I think the food is a notch above most Rosebuds (and I like Rosebud). I think Carmine's Vesuvio is a very good rendition -- might be one of my favorites.
I think the kitchen at Carmine's is just a bit more thoughtful and skilled than any Rosebud I've eaten at. The pastas especially seem to benefit from that. Always perfectly aldente -- good sauces, nice variety -- good to share.
Also -- if you eat downstairs there's sometimes someone at the piano -- and that's pretty fun. It's a nice room. Upstairs is sort of banquetty but it's still good.