Don't worry. There isn't a paid version. You have all the features
Well, kind of. Really the only similarity between the two is that they're both image editing programs made for Windows. Paint.NET is MUCH more powerful than Paint, but it's also MUCH simpler to use than Photoshop.
It's better in that it's got more features. And, once you get the hang of it, it's easier to do some things than it is in Paint. But with more features comes more complications, and that means there's a learning curve. Once you get the hang of it (Eli's advice above is excellent), you should find yourself whizzing through everything like a crazy person.
Layers are great once you understand them: think of them like a bunch of old-fashioned plastic transparencies all stacked on top of one another. Images on any of them can be edited, moved, and deleted without affecting the others, and the one on the top blocks all of the ones below it. Does that help?