Advantages of Debian over Ubuntu

Debian has some features that you could consider "advantages" depending on your needs and use cases.

• Stability. The Debian Stable branch has been tested extensively, generally for at least a year, as the Testing branch. The only updates Stable get are mission critical bug fixes and security fixes. This makes it an extremely stable platform (i.e., well-tested and little change).
• A tier-ed branch system for releases allowing you to pick the level of stability/up-to-dateness you need. Stable, Testing, and Unstable (plus backports, where select packages and libraries are ported from Testing to Stable). This provides a great deal of flexibility in how you decide to upgrade or stay with a certain version of a package or an entire release.
• The Debian Social Contract. A commitment to free software and the free software community. For the community, by the community.
• Debian is your way. You get a tremendous amount of choice and configuration options. There is no one "typical" Debian install. Debian is on your terms.
• Maturity - The Debian project has been around for a long time and is a stable part of the free and open source software ecosystem.
• Debian has been ported to many different hardware architectures. The current Stable release has 11 different ports. Ubuntu on the other hand is focused on the x86, and amd64 platforms.
• A LOT of packages. As in 29,000 worth. There's an old saying, if the project exists there's a .deb for it.

You'd have to further distinguish between Debian stable and testing/unstable, and between following all Ubuntu releases or only LTS releases.

• Debian stable and Ubuntu LTS release only every couple of years. Pro: you're not upgrading all the time. Con: the software and especially the drivers may get updated.
• Ubuntu has a few more things that work out of the box for inexperienced users, and a more polished recommended user interface. Debian is a little less beginner-friendly (fewer front-ends that hide the messy details) and a little more geek-friendly (fewer front-ends that hide the messy details).
• The core software (Ubuntu main) is more integrated. Once you go to universe, Debian is a little better because it's either more polished (stable) or more up-to-date (testing).

All in all, the difference isn't huge. I prefer to go with Debian stable on my machines, but recommend Ubuntu to others, and tend to use Ubuntu on newer hardware (especially laptops).

Advantages: More thorough testing, and structured release cycles. End-result, a more stable system.

Disadvantage: The stable archives are usually behind the latest version of software releases (including -dev libraries). This means you may need to manually install dependencies in order fill pre-reqs for that one-cool-program you really need the latest version of. Sometimes, you can work around this with debian-backports.