In a nutshell: you have 2 different set-ups.
Skate skiing has a motion similar to in-line skating. The skis are shorter and skinnier, and you need a groomed track. Skating is a heckuva a lot faster, and you need to skate up hills since the skis don't have scales on the bottom. Skate boots and bindings are specific to that discipline. They're made to be light weight and fast. The deal is you're stuck to skiing only where there are groomed trails, and it takes a bit more coordination.
Cross-country skiing covers anything from groomed trails (mini-trenches for the skis) to backcountry touring, and tele skiing on the far-end of the spectrum. You have a lot more variety in skis (waxless skis with scales/no edges, partial metal edges, waxless/metal edges, tele-skis with skins) depending on use. You can use system bindings like NNN BC or Salomon, or 3-pin bindings. XC is the way to go if you want to access the backcountry.