I would like to begin by saying that that scene from the Lord of the Rings is about as accurate a depiction of a medieval cavalry attack as James Bond is a depiction of modern espionage practices. Heavy cavalry charges were delivered in ordered ranks, generally two to three deep, the riders in line and under command of their sub-leaders, with plenty of space between lines to allow for room to maneuver in the event of something unexpected - say, a downed horse or a rider falling off. They would also be made at a considerably slower speed than generally depicted, both to preserve the horses (a horse with 250 pounds of armored man, saddle, and tack on his back cannot gallop indefinitely) and to preserve the riders' order. The goal was to break the enemy's line with the shock of the initial impact - or better still, to break their order by the fear of the charge - and if that failed, the first rank could draw off and allow the second to have a crack. Great mobs of close-packed horses did not mash themselves into the enemy like a scrum of rugby players.

This is from a previous post and deals specifically with Anglo-Norman cavalry organization:

The tactical building block of the Anglo-Norman heavy cavalry was a unit called a conroi. Each conroi contained around 15-30 knights, though it could be considerably less. In theory the conroi was composed of men from the same geographic area, very possibly servants of the same lord or landless young knights who had traveled to war to seek land and fame. This meant (again, in theory) that they had trained together for years and were thus quite used to each other's mannerisms and wholly capable of functioning as a cohesive tactical unit. We have pretty good evidence that they could execute quite complex maneuvers, such as the feigned retreat - not an easy thing to do in the face of the enemy. They generally went into battle in two to three widely spaced ranks, the idea being to give the second and third-most ranks room to maneuver to avoid a downed horse and the first rank room to withdraw in case of repulse or the aforementioned feigned retreat. Knights could and did dismount when the situation called for it - for example, if the ground was unsuited to mounted fighting or to stiffen the infantry. Our sources are far from clear, but my guess is they would remain with their conrois and be posted at the front, where their heavy armor and skill could tell.

Now, as to your questions!

so how common really were injuries due to a knight falling off his horse? Did a knight's armor really protect that well against a concussion or a broken neck?

Due to the paucity of sources, it's impossible to say how frequently certain injuries occurred. I think it can be safely said that medieval armor was not particularly suited to preventing the kind of injuries you ask about. Padding was worn under mail or plate, but I'm doubtful as to its effectiveness in these circumstances - certainly, I don't think it would prevent a serious neck injury. On the other hand, an armored man falling from horseback is not any more likely to be critically injured than an unarmored man doing the same. Provided he fell cleanly, there's no reason to think he would be completely hors de combat, though I imagine he would be very vulnerable while recovering from the shock of the fall.

However, we can certainly say that being a knight was a very dangerous job! Even excluding casualties of war, knights died or were maimed very frequently in hunting accidents or while participating in tournaments. Contrary to popular perception, for most of the Middle Ages the tournament took the form of a wild, chaotic mock battle fought over miles of countryside, at least sometimes using live weapons, and it's easy to see how injury could occur. Common sense tells us that some of these must have been ordinary mishaps - falls, dragging, etc.