Control group is fully manual. Fleet has some AI automation to make things a bit less micro-managey. When you make a fleet, one ship acts as the fleet leader. You then have, as an example, a "Fleet Cohesion" setting which determines how far away ships can stray from the fleet leader to engage targets (this works with the engagement radius setting). If you set it to "Tight" cohesion, ships won't move far at all, but if you set it to "Loose" they'll move more freely. The fleet also always keeps your ships in formation. When the ships are done executing their orders, they'll always fall back to the fleet leader.
To expand on this...
The Flagship becomes the only ship you really need to give orders to, to manage the fleet... if you prefer less micro. If you set the engagement radius of the fleet to hold position, the rest of your fleet will form up around the flagship and never break formation (unless you micro-order it to). At that point, you can order your flagship to engage whatever you like, and the fleet will stay in formation and engage it as well...
...to an extent. If the target of the flagship will get you the most bang for your buck out of the rest of the fleet, they will attack it. However, if your target is say, a LRM Frigate, and some strike craft fly by, your Flakk Frigates will detect that they could be doing more good else where, and fire at any strike craft in range... individual ships will attack whatever they have the best damage range against within range unless you manually tell them otherwise.
This is where the difference between control groups and Fleets comes into play. You can control group individual ships or groups of ships... For instance, you only need to order around the Flagship to get the fleet to move to where you want it to go, and engage enemies w/i range, or attack the Flagship's target. That said, recall that ships have a priority system on their targets built into their AI. If you wish to override that priority system, you have to issue orders to the ships in question.
So if you want, say, your fleet to move to a position near an enemy fleet, and target whatever they are best at, while your flagship (say a Kol), engages the enemy capitalship... just tell the Kol to attack the enemy cap. If, however, you want your carriers to hang back at the edge, and you want your bombers to hit enemy installations... you can make a control group of them. After you tell the flagship to do what you want, you can then hit the control group key to pull up your carriers, and have them stay put (micro'd orders > ai orders), and you could select your bomber control group and send them at structures... while the rest of the fleet follows the Flagship and engages based on bang-for-buck priority.
So really... control groups are just for micro. Also, if you give orders to any ships within the fleet specifically, they will break formation to do them, then return when done. If you want your fleet to stick together in attack, ordering only the flagship around is the easiest way... unless you want to micromanage the position of every ship in the fleet. NOTE: Any other engagment radius aside from "Hold Position" is essentially like giving the order "Break and attack". The other sizes just limit their aggression range.
As a last note, the formations ships fall into really do cover eachother pretty well... and I have found that keeping formation at the right times, can mean the difference between success and failure... If you can make a mess out of the opposing fleet by clever maneuvering, you've already taken a long step towards victory.
Oh, and sometimes it pays to break one gigantic fleet into multiple small fleets... for the reasons given in the above paragraph... YOU don't want to be the one being outflanked 
Hope this helps some,
-Itharus
