On Mouse Click (Tap) requires a touch/mouse down and a touch up on the same button. On Mouse Down (Touch Start) fires the event immediately on the touch down. Most OS buttons are typically using Click behavior, but sometimes the On Mouse Down behavior can make a document feel more responsive. With a click, if you have an accidental touch on the button you can drag to move off and the event will not fire, which is sometimes nice. It is really up to you!
(And as @Bendora brought up, using the Scene Swipe actions may offer a more natural iPad experience too)