I'm currently with the RAC, simply because I get a good deal with my car insurance. However I've been with the AA in the past and I can't really see much difference.
Back in my trade plating days, the firm I worked for had agreements with both the AA and the RAC, so either could be called out if a vehicle I was moving broke down. Over the 2-and-a-half years I did the job, I had to get one or the other called out loads of times. (The record was three callouts on the same day, all to different vehicles). Again I couldn't see any difference between the two organisations.
If you're looking for the best price though, consider Autonational Rescue. I used them for several years, without any problems when I needed them:
https://www.autonational.co.uk/