When you purchased Lion, what was actually downloaded was an application named "Install Mac OS X Lion.app". Running that application installed Lion, and then the app deleted itself. You're seeing the Download button lit because App Store notices that the app it downloaded has been deleted.
Same story with Mountain Lion, except the app is named "Install OS X Mountain Lion.app".
If the app didn't delete itself after installation, you'd eventually (in a few dozen years) wind up with dozens of huge "Install <some version of OS X>.app" apps in your /Applications folder, each of them taking up gigabytes of disk space, and none of them particularly useful. After all, they're all intended to be "use once and never again" applications.
Well, almost. There are scenarios where it's useful to have kept the installer around. The most common of these scenarios is that you want to install the shiny new OS on more than one machine, and/or on more than one partition. It's a long download that you might not relish repeating with every install. The second most common scenario is where you need to re-install the OS. While you can do that through the recovery partition, doing so would invoke another lengthy download. Faster to do it from a saved copy of the installer.
Those of us who planned ahead squirreled off a copy of each installer before running it. Those who didn't can click "Download" in the App Store to download again at their convenience, saving it this time. What that costs you is the time and bandwidth to download something you'll likely never have occasion to use, and the disk space to hold the copy; what it gains you is the flexibility to do the download on your schedule, not fate's. Your decision.
If you download Mountain Lion now, you'll get an installer that installs OS X 10.8.0. If you use it after 10.8.1 comes out, it'll still install 10.8.0 and you'll have to run Software Update (now part of App Store) to bring it up to date. That'll still be faster than downloading the whole 10.8.1 package.
If you download Mountain Lion after 10.8.1 comes out, you'll get an installer for 10.8.1. You'll never again have the opportunity to download a 10.8.0 installer. Not that you'd ever be likely to need one. (But in a pinch you could still roll your system back to 10.8.0 by restoring from backup.)