We have this discussion come up every once in a while, but it's always good to bring it up again for the new guys. Keep it civil, If you start attacking each other we will delete your comments.
There are obvious advantages for each. Reels are fantastic because you don't have that floatline dragging around you when you are diving and everything is streamlined on your gun. The only disadvantage is that reels can jam up or tangle or can get spooled which means you might have to drop the gun. The other disadvantage is that it is harder to put pressure on a big fish with a reel. If you are using a reel you might want a OSB bungee that Mori makes, it's a short bungee lined with dynema that stretches a lot, heres' the discussion thread: http://www.spearingforum.com/viewtopic.php?f=11&t=2494&p=25849&hilit=osb+bungee#p25849
Floatlines are great because you can put a lot of pressure on a fish, if you are doing breakaway you also get to detach the spear from the gun so if something happens you only lose the spear and not the gun. The disadvantage is they make a mess on the boat, are more difficult to carry on a shore dive and are prone to tangling when you are first getting in the water. You have a limited amount of line at 50-120' depending on what kind of fish you are hunting, and if you don't throw the gun over your shoulder or clip it off when you are fighting a fish it's easy to lose the gun. They are somewhat annoying as you can get them hitting your fins as you go under.
When I'm spearing seabass I use a 120' floatline, either a neptonics one that has an integrated bungee in front or a red triangle dark waters 120' skinny combo floatline which you can put pressure on and it stretches almost like a bungee. Both are backed by dynema core. Gannet also makes a really good one. When guys are first starting out or poor we always tell them to use a floating poly rope from home depot, they work fine but are a pain in the ass because they tangle up all the time and because they are rough on the outside will catch on your fins and make scratching sounds on the kelp. A good floatline is expensive will last a long time and doesn't tangle or tie up.
I'll try to take some pix of my floatline setups when I get home. I know ulusub has a big reel that holds a ton of line and the aussies have one too.