The main difference is simply the direction the shutter curtains travel across the film or sensor.

In practice, vertical-travel shutters usually have a shorter distance to cover than horizontal ones, so they can complete the exposure cycle faster. That’s why vertical shutters are associated with higher maximum shutter speeds and higher flash sync speeds.

Horizontal-travel focal-plane shutters were very common in older film cameras, especially fully mechanical SLRs. Their design fit well with classic film-camera mechanisms, but they generally had lower top shutter speeds and lower sync speeds than later vertical designs.

At shutter speeds faster than X-sync, a focal-plane shutter works as a moving slit rather than fully opening all at once. This means fast-moving subjects can show rolling-shutter-like distortion. The direction of that distortion depends on shutter travel:

• vertical-travel shutter: distortion is most apparent with horizontally moving subjects
• horizontal-travel shutter: distortion is most apparent with vertically moving subjects

So yes, SLRs with horizontal shutters definitely exist—many classic film SLRs used them. Modern DSLRs are generally vertical-travel designs because of the speed and sync advantages.