I use both the Canon 70-200s that the OP mentions and either would be a good choice, depending upon your needs. They are real workhorses.... top quality, durable, reliable.
The f2.8 lens is nice to have when you gotta have f2.8, is completely usable wide open, tho I tend to stop it down a bit for even better image quality when possible. This is probably my single, most-used lens and I've been using it for over ten years.
Recently I added the 70-200/4 IS as a backup and a lighter weight alternative. Its still not a small lens, but probably 25-30% smaller and lighter than the f2.8. I'd let that be your deciding factor... how heavy a lens do you want to carry. This will probably be one of those lenses you'll want handy most of the time, so figure on it being in your camera bag, if not on your camera.
The f2.8 lens comes with the tripod mount. The f4 is sold without it, but one is available at extra cost. The Canon tripod mounts are fairly expensive, but there are third party available for a lot less money. Mine has a Canon t-pod mount on it... don't know how that compares for fit or quality, with the 3rd party.
The f2.8 lens uses and comes with an effective, "tulip-chaped" lens hood. The f4 lens also comes with a lens hood, tho it isn't "tulip-shaped" (I've seen that type offered by 3rd party manufacturers, if preferred).
The f2.8 lens uses 77mm filters. The f4 lens uses 67mm.
Eventually I'll add or upgrade to the 70-200/2.8 IS II, but I'm not in a rush to do so. The original f2.8 IS is pretty darned good.
Haven't tried the third party lenses.