Well, for your longer flat rides a cross-country bike would be better because it has less travel on the suspension, so you don't waste as much energy with your bike bouncing from having TO much travel. However a lot of bikes you can lock out the suspension, or add air to it so it's stiffer.
All-Mountain would be better for rough riding. So really any kind would be fine. I ride a cross country, and have taken it on some Very rough trails, it does fine.
Here's links to pictures of some of my favorite trails I ride with a cross-country bike...
I posted all those pics of south saint vrain :)
Its all a personal preference, my cross-country bike works just fine on these trails, I've also seen plenty of people using a hard tail on these trails, but an all-mountain would make things easier in some sections, and would be a lot smoother ride overall. All-mountain bikes do cost a bit more,
HEres Konas full-suspension cross country bike followed by their all mountain bike..
Oh another difference is sometimes the cross-country bikes will have a more flat top bar, and all-mountain bikes will have a more steep angle of the top bar going from the seat post to the handlebars, and you'll notice the front tire/front fort stick out more on an all-mountain bike, that steeper angle makes it a lot easier to go over steep/rough terrain.