I went cage diving with GW in South Africa last year. I went to "Shark Alley/Junction" the place they do all those Discovery Channel videos on GW sharks. It was pretty cool in many ways but also a let down.
Cool: dude, it's a great white shark!
Let down: They've made it more like an amusement ride than a big adventure.
Now admittedly I've done quite a lot of diving in my past. I've been an instructor for 20 years now and dived all over the world including the Galapagos (that's the best place btw) as well as deep restricted wrecks like the USS Monitor. So I'll say up front that I'm pretty spoiled for big diving adventures.
Here is a rundown of how they did it in South Africa:
They take you and 10 others out to a general area where there are GW sharks. There's usually 2-3 other similar boats out there too. They start chumming the water and when one comes around, all the boats gather around.
Then they put you in the cage. The cage is tied to the side of the boat so you just go from boat to cage- no swimming in the water. You aren't diving, you are snorkeling. BTW, the water is fairly cold- low 60's I'd say. Unless you bring your own wet suit, you'll get one of theirs which inevitably doesn't fit very well so you get chilled pretty quick.
While you are in the cage, the GW shark is making passes. They are big animals and pretty impressive underwater. On the surface, someone is luring the shark in with a tuna head on a rope. Basically the throw the tuna head out and as the shark lunges for it, they pull it away. Think of a dog jumping for a treat.
Unfortunately, that's pretty much how it goes. You rotate in/out of the cage so that everyone gets enough time in. You can go back later but generally speaking the cold saps people's desire to stay. The shark trying to chomp on the tuna is cool as you see all the teeth and sharky stuff but after a while you get too cold to stay.
Sorry, hope I'm not being a downer on this. It was a lifelong desire to cage dive with GW sharks. I don't regret having done it one bit. However on the flip side, I probably wouldn't go do it again.