In the US, "in a hurry" and "in a rush" mean almost exactly the same thing. They are equally common.
"Bit of" is an example of understatement. Literally, "I am in a bit of a hurry" means "I am slightly hurried." In real life, depending on the speaker's personality and tone of voice, it can mean just what it says. Or, it can mean the opposite. It can mean that speaker is really in a very big hurry, but is calling it a "bit of a hurry" to sound sound relaxed, calm, and polite.
"Hurry" means that you are doing things very quickly, under time pressure. "I have too many things to do and not enough time to do them in. I'm in a hurry." It means you are running around frantically doing lots of things. "Rush" can mean exactly the same thing.
"Rush" sometimes carries the extra idea of one specific thing that must be done by a deadline. "I'm in a rush to get this packed for shipping before the pickup time." In the United States, a "rush job" is a job that must be completed faster than usual. "Don't rush me" means "I'm working as fast as I can, don't try to make me work any faster." Someone might say "That paint job doesn't look very good, but what can you expect? It was rushed."