I'm not in a position to list all of the different collar types. I don't know why that would be useful knowledge.
But to answer your question of pros/cons:
They all serve the same fundamental purpose. Namely to hold the plates in place and provide a balanced bar.
In regular commercial gyms, you'll probably find the spring clips. You just grip it so the spring opens a bit, slide it on, and release.
Pros: Easy and intuitive to use. Will hold pretty much any weight regular gym-goers will ever use. Weighs next to nothing, so they don't have to be accounted for when figuring out how much to put on the bar.
Cons: Can not be counted on to hold the weight thrown around by professional powerlifters and olympic lifters.
Then there's the bigger collars like the ones we see in your screenshot. It's a sturdier type that slides on, and is fastened by screwing a clamp in place that fastens the collar's grip on the bar.
Pros: Will hold any weight liftable by any human (to date).
Cons: Takes a bit more time to fasten. Weighs a non-trivial amount, and will have to be calculated in as if it were a small plate.
Both types come in a variety of shapes and sizes. There's a plethora of manufacturers, and they all have their own design.