Mint is way, way closer to Ubuntu than Ubuntu is to Debian. I will try to explain without getting too geeky:
Debian has three main branches of development: unstable, testing and stable. The stable release (currently 5.03, named "Lenny") is the currently released one. "Unstable" is named poorly, as it does NOT mean the software is unstable, it means that the system itself *changes* daily (in contrast, "stable" gets only critical security updates, which are rare). Brand new versions of software in the Debian system are dropped on day 1 into unstable. Once the teams decide on the version they want to develop for the system, it is moved to "testing", where the source code is examined, it is compiled from source, configured and re-configured, and the system tested extensively. Once the system has been deemed ready to rock (in Debian's history, that's about every two years or so), the whole shootin' match is moved from "testing" to "stable" and BAM, you have yourself a new release. You are free to run systems based on unstable, testing and stable; each with it's own advantages and disadvantages. "Unstable", for example, has brand spankin' new versions of everything, but might not be refined and dependable, and the system may change quite a bit from day to day! "Stable" is rock solid, but has very old versions of software. I use "stable" for friends and family, and run "testing" at home.
What Ubuntu does is take software from the "unstable" branch (meaning the software in there is up-to-date and fresh) and they begin developing, testing, configuring and re-testing the software independently from there, and largely independent of the Debian developers. Ubuntu has a clockwork release cycle of 6 months, whereas Debian releases "when it's ready". Ubuntu, then, can diverge quite a bit from Debian on the same piece of software, depending on how they choose to compile and configure it, and which version they use in the greater system.
Mint, on the other hand, just takes a stock Ubuntu system and adds stuff that Ubuntu cannot do legally, like distribute codecs and proprietary drivers. They also modify the look and feel, and I *think* they add some handy configuration tools.
As far as servers go, Debian versus Ubuntu is the same as in the desktop world: Debian is conservative and more adherent to free software usage (Debian doesn't use, technically, the "Firefox" browser, as Mozilla has some type of license on the *brand* that Debian doesn't like, so they re-compile it from source and call it "IceWeasel". They take the "freedom" dimension of "free software" pretty seriously), whereas Ubuntu is out there making corporate partnerships, getting hype and press and using proprietary, and usually newer software. Ubuntu is easier to administer and set up, but Debian is more stable and solid. Debian assumes you know your stuff to an extent, while Ubuntu is more user friendly. Personally, I think Ubuntu's place is as a new user's introduction to desktop Linux, while Debian's place is for advanced users, servers and for others who wish to build their own OS: Remember, it's just not Ubuntu that's based on Debian; lots of others make Debian-based distro's for one reason or another.
