Oh man! Where to begin. There must be countless threads covering this, but I'll do a quick and dirty.
A reel, assuming that its big enough and with the right size line, lets you have far more line than most people would be willing to tow around as a float line. But if all that line is taken by a fish, then you may lose the gun. A reel makes it a lot easier to get in and out of the water as opposed to deploying a float line, coiling it back into the boat, etc.
A float line, if rigged as breakaway, lets you keep the gun safe in your hand while the flsh is connected straight to the float line. If you have a large enough float, the fish can't take it under. But even if the shaft, shooting line and float line are all taken, you still have the gun in your hand.
Having the gun in your hand is a mixed blessing. If its attached to a reel line, then you can just release it, perhaps while diving to get a fish out of the rocks or kelp, and know its not going to float away.
If its not attached, you have to keep track of it. One standard is to stick your arm through the bands and shoulder the gun. That's OK in open water, but I don't like to wear a gun when I'm trying to get down through thick CA kelp. Another option is to clip it to the rear end of the float line.
I think I'll stop for now and wait for input from others and/or more questions.