I haven't used Fedora since version 2, and they called it Fedora Core back then. I thought 3 and 4 went downhill. I really don't like RPM and RPM-based distros, but I haven't used any in a while, so they may have gotten better.
So, I can't really give you a comparison between Fedora 7, 8, and 9. I will echo runswindows95, though. I believe Fedora is on a six month cycle between releases (might be mistaken), and during those six months, a lot can be done. Each new version includes newer versions of the Linux kernel, X server, desktop environment, and so on. Bugs and quirks will be fixed. Generally, only security issues are fixed in older releases, and only during the time the older versions are supported. Afterward, it's up to the community to fix those problems.
If you are on ancient hardware, you might want to look up the system requirements for each version of Fedora. If the computer is really that ancient, then Fedora 7 may have slightly lower system requirements; however, I doubt you'll need to take this measure.
If you don't have any special circumstances, run the latest version of the distro. It will likely be the most compatible and feature-filled choice.