What are the differences between the demo and full versions?
In a nutshell, the full version has a persistent game world, meaning anything you launch will saved and resumed later, which makes it possible for you to have multiple missions happening at the same time.
Also, we've done a massive overhaul on the terrain system. No more visible quad seams, and it's smoother overall.
That was the 0.14 update, on this latest one, we added the Spaceplane Hangar, where you construct, well, space planes, on a horizontal construction facility, with mirror symmetry (you're also able to rotate parts manually now also).
There is now a new patched conics trajectory prediction system. It will calculate, based on your orbital parameters, whether and when (and how) your trajectory will encounter another sphere of influence, or escape the current one, and display your new orbit around it.
And lastly, we added a new moon around Kerbin, called Minmus. This small moon is not much more than a captured piece of rock, but it does have enough gravity to give you two SOIs inside Kerbin space, and a place to practice interplanetary transfers on a small scale, where you don't need to spend massive amounts of dV to make big changes to your orbit. (In fact, I heard a popular technique for reaching Minmus is to match its orbit, than burn retro for a full 180, and meet up with it going the other way)
That's about it really. There are more changes and tweaks, but I think Pipcard's links should cover that.
Cheers