Big-mountain skiing developed from "extreme skiing" in the late '80s to early '90s. As the name implies, it involves descending large, steep mountain faces at high speeds, and it is usually done in the backcountry on ungroomed snow (often powder). Any line at all can be taken down the mountain, and skiers often air off of cliffs, windlips, spines, etc. It is not about competing or trying to race to the bottom. It's about creativity, having fun, and enjoying the snow.
Newschool skiing is done in terrain parks at ski resorts. Largely inspired by freestyle snowboarding, it involves doing tricks, hitting jumps, sliding rails, jibbing, and skiing halfpipe. It is becoming increasingly popular among youth, and many people are pushing to have halfpipe skiing included in the 2010 winter olympics, which would do a lot for legitimizing freeskiing.
Most people still think of conservative, nerdy golfers when they think of skiing, because freeskiing has not had a lot of mainstream exposure. these are the same people who still believe in the feud between skiers and snowboarders (which no longer exists), with snowboarders being the irreverent youth and skiers being the elitist pricks on sticks. This stereotype is outdated, and soon the rest of the world will catch on to that.
Other Snowboarder: Educate yourself, bro. He's a freeskier. That's his sport.
Snowboarder: Oh. I totally understand now. I totally respect his sport, 'n shit.
Freeskier: Cool. I respect snowboarding because it's done so much for freeskiing, 'n shit.
Other Freeskier: Let's all go ski and ride and get high 'n shit.