There will be virtually no difference. Both will be completely stable in the fact that you will be getting NO new software for either, if you are using a released version of either Ubuntu of Debian. You will get only security patches to the old versions or other trivial patches to the old version of the software which it was released with and not new versions. This will not be different for either Debian or Ubuntu, the policies here are the same for both. And this is what makes either choice "stable". One misconception about Debian 'stable' is the meaning of stable. When we label a release as "stable", we do not mean stable as in "not crashing" we mean stable as in "not changing".
One difference will be the life cycle. Ubuntu LTS releases get 5 years of support. The most recent LTS release, 12.04 was last month. It will be supported until 2017. This means that you could install it now, and still be running the same 2012 versions of software in 2017 (but you will get security support for them).
The most recent version of debian was released in February 2011. The next version of debian is about to be "frozen" in order to stabilize it for a release. There is no reliable way to determine when this will actually be released. This is one major difference between debian and ubuntu. Debian will make the next release when this frozen release is virtually free of known release critical bugs. At this point, we have about 700 more to close, as can be seen here. Ubuntu, in contrast, releases on a predictable schedule, regardless of the number of known bugs.
Once Debian makes a release, it will support the previous release for one additional year. If you were to make wild ass guess about when the next Debian release (7.0) will be made, you might randomly guess "December 2012", that would mean that the current release (6.0) would be supported through December 2013, and you'd have until then to upgrade to 7.0. You could make even wilder crazier guess about when the following release would be, and guess "about 2 years" which would mean you would then have until about December 2015 to upgrade from 7.0 to 8.0.