I'll leave it to others to suggest a good operator, but I can describe the diving.
Jax diving tends to be long boat rides, b/c many of the sites are 20 - 40 miles offshore. Think most of the day vs a few hours, as in the Keys.
Profile is ledges, and wrecks. Because you are out 20+ miles in the open Atlantic Ocean, conditions can be rough, or turn that way while you are out (or not, as the case may be).
Many who dive Jax are spearfishermen, so charters cater to this activity.
Those who do it regularly describe the diving in glowing terms - its been a while for me and I don't spearfish.
Really depends on you - most times folks coming down to squeeze a dive or 2 in are looking more for the Keys experience in terms of time but by all means give it a shot if this kind of diving appeals to you. The Keys its not!
Also, do a search - this topic was recently discussed and there were some shop recommendations made.