2 strokes generally have way better power-to-weight ratio. They're considerably less fuel efficient, tend to be noisier and less smooth than 4 strokes. 4 strokes are much more complex and expensive. They generally have a more linear power curve whereas 2 strokes are usually set up to have a region where they make a lot more power, this is the "power band" people refer to. 2 strokes can be set up to be reasonably linear though - look at outboard boat motors as a good example.
2 strokes also burn oil by design, either as pre-mix or through an oil injection system. This is because the bottom end and bore is lubricated by the fuel mixture instead of a dedicated sump, meaning oiling and lubrication is vastly simplified, and you can usually run 2 strokes on all sorts of crazy angles without them freaking out and dumping oil everywhere like a 4 stroke would. This makes them well suited to things like chainsaws and line trimmers.
4 strokes use a valvetrain to control the inlet and exhaust from the combustion chamber. 2 strokes use a porting system on the cylinder wall which, for at least part of the stroke, has both the inlet port and exhaust port open simultaneously, resulting in a quantity of unburnt fuel mixture being scavenged with the exhaust gas.
Skiing equivalent... I would say it's like rocker vs. camber - they each have a time and a place where one is better placed than the other, and you can't necessarily say that one is always the best.
tl;dr: The third poll option.