I'd say the main and most significant difference is that Royal ships have more to do in terms of fun and unique activities, such as rock climbing, zip-lining, mini golf, etc depending on the ship. On the other hand, I'd say Celebrity ships are lower in activity level, so things to do are eat, watch, sit, drink. So, really depends on your own activity level and how you like to relax! 🙂
In terms of crowding, we usually go during shoulder season so we avoid most of the kiddies. Should also help with space in the pools and such. There is often low traffic areas as well depending on the popular activity at the time (check the activity schedule/cruise compass).
Service on both are equally lovely and attentive - I personally did not find a difference!
Just my two cents. Hope this helps. 🙂