Aqaba has better dive sites- most of them have a more interesting topography (i.e. steeper slopes, a few real "walls", bigger wreck, more pinnacles, bigger gorgon corals etc. etc.). You may occasionally meet a turtle, a few Napoleon wrasses, etc. Easy shore diving, guided only (AFAIK).
I don't dive in Aqaba frequently enough throughout the year to tell you about seasonal status of fish schools, where turtles sleep and so on.
In Eilat there are a few sites with a steep slope (almost a wall
) going from ~15-50m, a nice wreck (decommissioned missile boat), a few sites have huge schools of yellowtail barracudas, you may chance to meet turtles, sea horses, frog fishes, eagle rays etc. Actually Eilat compensates for the lousy amount of "big game" (and relatively poorer reef) with a real big diversity of fish species to be met during a dive, the easiness of diving and of course- no guided dives are compulsory as it is claimed in Egypt and Jordan.
Sure, in both places (actually everywhere) one needs some luck to see real nice stuff- as a tourist coming for a very brief period (on a "business trip") one doesn't know every corner of the reef and where specific fish lives. In this case, you'll get a better impression if diving in Aqaba, or Dahab.
To summarize, my recommendation is (assuming you are coming from Tel Aviv):
If you can spare only a single day of diving: Eilat. Any other option would be extremely tiring!
2 days: Eilat or Aqaba (you choose
)3-4 days: Dahab, or perhaps Taba-Dahab safari, can be organized through Israeli operators such as http://www.seaqueen.co.il. Another option can combine 2 days in Aqaba and a day to Petra and the such.
5 days or more: Sharm-based safari. You can organize through the mentioned above, who can perhaps take care of all your transportation needs from TLV, or any of the PM's you'll probably get with invitations :cool2:
