it's not the edge of the earth, but you can see it from here
A weymouth is within the subset of curb. Curb bits have leverage and a curb chain/strap to apply that leverage.
A weymouth is an 'english' curb bit designed to be used with a bradoon.
Fixed cheek is more stable, with no movement. Some people call a sliding cheek the cheek that turns... there are also sliding cheeks that have a little tiny bit of play up and down.
Just like with the snaffle, some horses prefer a more fixed position, some do better with a little play. Ironically, head tilting can be helped by either.
Traditionally, a fixed cheek is more a one-handed bit (or three-and-one) and a loose cheek is for two hands. No one really pays much attention to that any more. When you use one rein on a fixed cheek, in anything but an up or back motion, (i.e. opening, indirect) the bit will actually tilt a bit in the mouth. You can use a tiny bit more of lateral motion with a sliding/loose cheek.
In western riding you learn the difference in feel between the two--because you are always riding one hand on the curb. I think it's a subtlety that is often lost for english riders.
In my experience, a sliding cheek can provide a little more of a buffer between hand and mouth - we used to use them on our saddle seat horses with youngsters or with ammies that didn't have great hands. The little extra lag time between the hand action and the bit can keep a horse more comfortable. I think a reasonable analogy is that of a car that has sport type steering vs. a cushy luxury car type steering - one is very immediate to respond, and one has some more give and more play. The sport steering doesn't give you as much room for error, but the cushy luxury steering is softer and less quick to respond - more forgiving though it still gets the job done = )
Others might have different experiences, but I like them on certain horse/rider combos for that reason.