You can have different backgrounds on different Spaces. It's great to be able to organize a little mental model of your workspace by using differently colored backgrounds. I find I'm using the swipe gesture to change spaces frequently, and it can get a bit disorienting; every space looks almost the same except for a slightly different arrangement of rectangles.
That's true; although the desktop background pane in System Preferences can be quite buggy, to the point where it's far from obvious that it would even be possible to make the backgrounds different.
For instance, it will only appear by default on the displays you have on the desktop you're using, and it shows the background of that desktop in its mini-rendition even if you try to switch desktops. It can also sometimes show the wrong background in the preferences pane even after you've changed it. So although it does work, it hardly gives the user confidence.
The most reliable method seems to be to use the Finder's "Change Desktop Background..." contextual menu from the desired desktop, to force System Preferences to open in the right state.
