What’s the difference between using the materials button vs. the texture button to set up nodes for an object? I can understand why there is also a compositing button, and I would use that on the render result normally.
They’re basically the same thing. Materials and Textures buttons are just another way of setting up nodes and connections. Nodes give you more control of course, but the buttons might be easier to set up the overall materials before touching nodes.
The node editor gives you much more scope once you wrap your head around it. IT seems daunting at first, especially when you see some massively complex node trees, but if you start with a basic mix shader in the ubttons and then ‘use nodes’ in the node editor, it gives you a bit of an idea. To be honest, I can set up a basic mix far faster in the node editor than I can with the buttons. With the Node Wrangler AddON active, it is very quick to get into. 
I think my question was not clear. I am talking about the node editor. I know that using nodes gives much more control over materials. But I am referring to the panel in the node editor window where you can choose which node tree type to display in the editor window: shader (the little sphere), compositing (the two overlapping squares) or texture (the checkerboard icon).
Ahh, I get you now, the texture is for setting things up like textures in Sculpt or such rather than an object material. The only down side to that is, I’ve found using image textures in those nodes doesn’t work very well, only procedurals. I think this is a known issue as when I was googling it, it seems other have issues with images in texture nodes, (things like using a colour ramp on an image to tweak it for sculpting).