Thing is, horses are a lot more agile than humans, and can literally turn on the spot a lot faster than somebody running around it. The guy on top of the horse, can still block / parry, and is quite high up, so it is technically harder for him / her to be hit. Yes horses would be stabbed instead, but that would give the opportunity for the rider to swing at the attacker. Horse itself, would not just stand on the spot either.
So basically the equation is: horse = more maneurable than human, rider is on top of the horse, rider can still turn in the torso (to an extent), rider = higher than unmounted troop, horse = heavier than anyone on the ground and can literally push through a tight situation, horses will take a lot more than a human to be knocked down. Therefore it would be safe to say that the guy on top would have an advantage, providing they know what they are doing.
Obviously I won't deny the fact that a horse which doesn't trust the rider, with a crappy rider on top, would not work too well since the horse would panic, throw the guy off and try to escape. However even if the horse is not 'trained for war', but can trust the guy/gal on its back, it would be a completely different story. Even if the horse is scared, he would attempt to defend himself and the rider.
Obviously I won't deny either that long spears and such were used to counter cavalry. (thats why they were invented in the first place). This actually just proves my point, that with sword vs sword combat, a mounted troop has the advantage. Spears would be used to stab the horse I believe, since the rider can still dodge / deflect the spear coming at them, since the only sharp point is at the tip of the spear. This is why spears are not that useful against infantry, and spearmen would not work too well (in reality) on their own vs enemy of similar numbers. It all comes down to strategy.
Its a bit hard and pointless to explain this to people who only seen horses in a computer game tough, and in most games today horses = an increase in speed, and nothing else. So I don't blame anyone for not understanding.