The key difference is the position of the handle when the clamp is locked. A horizontal clamp has the handle position horizontally (parallel with the base) when the clamp is locked. A vertical clamp has the handle positioned vertically (perpendicular to the base) when clamped. The intended use you have will decide which you should buy, but be ware that some vertical clamps have a slightly smaller opening range than horizontal ones. Check the specs!
You should prefer vertical clamps if you are making a jig to be used on a table saw, because the horizontal clamps might have their handles interfere with the fence on the saw.
Conversely, if you were going to use the clams on an assembly table, you might prefer the horizontal clamps, because the vertical ones might get in the way by sticking up. (Also, vertical might interfere with a scroll saw, while horizontal might interfere with a drill press or miter saw.)
You can mount the horizontal clamp in a vertical position or the vertical clamp in the horizontal position.
A horizontal clamp closes or applies pressure in the flat position or with the handle horizontal to the clamp. Vertical clamps when closed the handle points up to the sky.