A couple things: first, because the horse is physically in the same space as the archer, in most cases this limits the size of the bow in question (hence why most horse archers used some sort of recurve bow to gain more power from a shorter bow) so you can’t shoot as far as a bow that’s as tall as the person wielding it.
But second and more importantly, an archer on the ground only has to focus on one thing: shooting as fast and as accurately as they can. A horse archer has to focus on riding the horse and shooting, which makes it more difficult to concentrate.
Third, shooting while in motion is inherently more difficult than on the ground, because both you and your target are now in motion which means you have to make constant adjustments to your aim, and it’s further complicated by the fact that the gait of a horse means you’re dealing with both horizontal and vertical movement.
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