Mounted Crossbowmen

As to why someone would use mounted crossbows versus mounted archers, the qualities of the crossbow themselves make it readily apparent: 

1) crossbows require minimal training to use compared to a longbow;

2) the trajectory of a crossbow is basically flat, while a bow needs to fire in a high arch if the shot is to cover any significant distance.  The crossbow is more or less a point and shoot type of weapon (see also #1 above);

3) the flat trajectory and firepower of a crossbow bolt allows it to be used more effectively in bad weather conditions where bows and arrows would be rendered useless;

4) bolts could be loaded and held at the ready indefinitely.  A nocked arrow can only be held for just so long

Even if a crossbow designed for mounted use has to be small enough to be no stronger than a bow, thus giving no special advantage for range or penetration, the above to qualities could be enough to justify the existence of such a troop.

And because wikipedia is such an awesomely reliable source  it appears the Germans used mounted crossbow troops to fire a single volley at close range before drawing swords and charging.  While a short range volley like that might be pretty effective in terms of dealing damage, since it's a single shot volley, the effect was presumably more psychological than anything else.