Well, the main difference is that they typically acclimatise over a longer period of time. It's not all about increasing Haemoglobin levels, as even a few days help considerably. And Sherpas/Andean populations have genetically adapted to live at altitude so even though there are people who can climb to 20,000 feet this probably isn't something just anyone can do.
A climbing friend of mine got altitude sickness at just 3000 metres - short holiday; drove most of the way up the mountain and walked the rest on day 1.