Occasionally I've seen skiers free-heeling it down the slopes in telemark style but not understood why they do it. Is the main benefit for exercise, style, control or just to try something different?
Telemark -- where the toe is attached, and the heel is free to rise up and down -- allow skiers to skin up back-country slopes with a more natural and efficient stride. However, AT (or randonee) rigs allow skiers that ability while doing alpine turns on the way down.
So, in this day and age, why do people still Telemark:
In short, yes to all points in your question. It is different, requires more control, has more room for style, and is a hell of workout. But beyond that, and beyond back-country access, it is pure, unadulterated, fun.
"Free your heels and your mind will follow."
Telemark skiis are much more suitable for cross-country skiing than Alpine skis. Having the connection only at the front of the boot allows you to walk and cover long distances very efficiently across flat snow.
But even in cross country-skiing, you will sometimes need to ski downhill, so the style of skiing with one dropped knee on corners was developed to allow for controllable steering. You can't easily ski normal downhill style with a free-floating heel.
So people with Telemark skis will do this for practice, and for fun.
The main (and classical) advantage of telemark skiing is not when you're going downhill, but when you're going uphill.
The free heel (and a set of skins) allows the telemark skiier to skin up a hill much like in ski-touring, cross-country skiing, or even snow shoeing. Telemark skiing is very similar to ski-touring except that in ski-touring you lock your heel back in place when you're about to go downhill.
Telemark skiing goes back to the days before chairlifts, when the ski was a floatation device used for ascending as well as descending.
As for the fancy telemark turn (with the big kneel/dip at each turn), it's because the soft telemark boot can't efficiently transfer the force of a rigid modern ski boot, so the dip helps transfer some of that.