I work in Sharm and although I have never dived in Aqaba we have a centre there which a good friend managed for a while, so in terms of the major differences:
The reef in Sharm is the best in the northern Red Sea, I don't think this can be disputed but the downsides are that they can get very busy, and are mostly only accessible by boat, which some people don't like and Ras Mohamed and Tiran are full-day trips 8am - 5pm which can be problematic for non-diving family. Aqaba has much less diving tourism, so it might be a bit more relaxed
Aqaba is a little more expensive than Sharm - my own company has centres in both locations and our Sharm prices are cheaper.
Sharm is not a real place - it's a tourist resort. Cool places to visit (including Aqaba!) are readily available and everything is well catered to in a very touristy sort of way. Aqaba is a real place with real history and is a pretty new tourist destination for Europeans, especially divers. For a lot of people, that makes a difference in location.
Finally - I would visit diveaqaba just to meet the Danish girls!
Please say hello to them from me btw, if anybody visits. Like they have any idea who I am, of course, which is to say none at all.
Wherever you go enjoy your stay!
C.
