I generally stick with music that lies outside of the first-world-white-peoples' consumer orgy that is the 'Rock-n-Roll Hall of Fame' set of bands (and especially fucking stadium rock acts from the 70s forward like both of these examples), but if I were forced to make a decision, I'd take Queen hands down. Songs like 'You're My Best Friend', 'A Kind Of Magic', 'Under Pressure', 'Somebody to Love', 'Killer Queen', 'Bicycle Race', etc... are simultaneously catchier, more stylistically versatile, more boldly adventurous, and more virtuosic than anything Rush ever did, which is saying a lot, because Rush at their best (i.e. in my opinion, roughly Moving Pictures through Counterparts) were pretty fucking amazing.
Thinking on it in a different way, I'd posit that Rush put together better whole records while Queen absolutely annihilates them on the basis of a 'greatest hits' criterion. It should be noted that I've probably only heard about half of Queen's full records and that a lot of the non-hits don't really do much for me. Still, the hits hit hard!
To be sure, I say all of this as a bassist of 20 years who spent a good half of that time worshiping Geddy Lee's playing and transcribing Rush's songs. I still enjoy all of those great Rush records from the 80s, but the total musician in me ultimately finds their music a bit too boxy, formulaic, pretty corny, and overall, a bit too 'provincial' in terms of musical language and overall group aesthetic. I'd unhesitatingly take them over most other prog rock bands (i.e. later groups from that subgenre like Dream Theater and Porcupine Tree still aren't worthy enough to lick Rush's boots...).
EDIT: In all honesty, I'd probably take the best work of Genesis and Phil Collins over both of these acts. I go back to their best songs more than anything else from that era.