I have had drinks at both places, but not food in the pubs. I have had a meal in the RC dining room.
As for pub atmosphere. RC is darker and smaller. RC has very few tables, and the bar is a stand-up bar, without bar stools. They do have a piano player at times. RR is a lot larger, brighter. There are a lot of tables, as well as barstools. There is nightly entertainment, though the evening I was there I would not consider it 'Irish Pub Music.' It was more like covers of every popular British/Irish/Scottish/Welsch singers out there. They are both fun, but nether are a real English or Irish pub experience, especially if you have been to those countries.