The physiological difference is fast and slow twitch muscle fibres composition between riders.
In cycling, its the Ectomorph versus the Mesomorph. (google these terms for more info if you like)

If your a pro cyclist, chances are you have a large number of slow twitch muscles as it is, but the degree to which you have them differ from rider to rider. Track cyclists for example are Mesomorphs, a lot of fast twitch muscles, and it shows, they are HUGE.
Grand Tour cyclist are broken into 4 categories,
All rounders,
Climbers,
Sprinters,
and Time Trialists

The all rounders make up the main GC guys, as well as the domestiques. The GC guys are just a lot more talented. The rest are self explanatory.

To target your question a little more,
its more about power to weight ratio and psychological. Some proof is illustrated by the fact that some riders have started out as time trialists for example, or sprinters, and over years develop into somewhat of climbers.
Lance Armstrong for example, had a career before cancer, and he was not at all that brilliant a climber, he did his fair share, but was more of a "one day guy". As the propaganda would tell you, and I bring it up because it has some convenient truth to it, is that after cancer he came back a lot lighter than before, while maintaining the good part, nice strong legs. This meant his power to weight ratio was immensely better than before cancer, allowing him to stick it out with the climbers, and sometimes, even drop them.

Compare a man like Carlos Sastre to Cancellera. Carlos is a weaker man, in a race of truth (time trial), he can lose minutes on end, but when the roads turn up, advantage Sastre, because the power to weight ratio swings to his favor.

Drafting, well, it helps very slightly in mountains if at all, but barely, or should I more accurately say, the power to weight ratio thing FAR out-wieghes any mathematical advantage of drafting, (for the American readers, im not saying "there is no drafting advantage")
Obviously too, the speeds dont allow for the advantage of drafting to make a big difference.
Here is an interesting point though.
Drafting someone in the high mountains does make a difference. Contradictory? No. Because psychologically it makes a world of difference.
Proof in the pudding as an example? You know good 'ol "Big George", a man for the classics; a good all rounder. Well, a few years ago, he paced Lance up climbs, dropping and putting climbers into difficulty!! The reason is that he knew he was carrying his leader in yellow, and psychologically that gave him the will power to burn his legs and not give up, even though he really isnt meant to be a climber. So yes, team work DOES make a difference in the mountains.
Your logic about powerful riders, like sprinters has slight merit. I can use Jan Ulrich as an illustration. He had some VERY POWERFUL legs. He used his slow, powerful cadence and just deasel powered through time trials, and snapped climbers legs. He could not respond to attacks, or any explosive moves in the mountains, his power to weight ratio was against him. When climbers would attack, he just sat there and didnt even try to stand up for a violent reaction. He just used his big gear and turned it nice and steady, but ultimately, the lighter rider's power weight ratio is just too much to beat.
Realistically speaking though, psychology aside, power to weight ratio, bottom line, to answer your question.

As to why the mountains separate boys from men, well thats an easy one.

In the mountains, there really isnt a "main field" that finishes together. (apart from the groupettos, these are the groups of sprinters that finsh almost 30min behind, they just form little chit-chat groups and ride to the finish and try to make it within the cut-off elimination time, which is a certain percentage of the day's winner's time, not limited to the mountains, they have it on all stages. They just chat about girls all day and enjoy the sunshine as best as they can) So simply put, its each man for himself, and hence, each man starts to get his own time that is added up to see where he stands overall.
You will notice that in the last 3 stages that were flat, The yellow jersey and Lance position pretty much havent changed positions. (exclude the time trial). When Lance was 10th at the start of the stage, at the end of the day, he was still 10th. When he started the day 0.0 second behind yellow, he was still 0.0 second behind yellow. And im sure as you know, its because the main field more or less finishes together in a group on a flat stage, and in fact is given the same time.

Some detail on that if you are wondering what that is all about.
In fact the same time is given to everyone in the main field within the last 3km.
(if your in a second group a little behind, everyone gets the same time in that second group that separate from the first group and so on, its a group time you get) So for example, if you have 4 days in succession of flat stages, and it all group sprints, the overall GC times dont change at all. So the boys and the men stay together.
Once the road turns uphill, each man gets a separate time at the finish line so long as a second of day light is between your wheels. If you finish 3 seconds behind someone, you are not given the same time as you would in flat stages.
As you can imagine, pretty much everyone gets a different time, and so, at the end of the day, if you were tie with the yellow jersey at the beginning of the day, and you finished 3 seconds behind him, you are now 3 seconds behind overall, where as on a flat stage, if you finished 3 seconds behind the group at the line (in the group you were actually in within 3km to the finish), you get the same time. Basically, there arent much changes to people's times in flat stages, and in mountain stages pretty much everyone gets different time. So it becomes " a real race" in the mountains. And of course, the stronger man wins, provided he can climb, LOL

A little long winded and more info tan you would care for, but there it is.