Debian: Generally going to be considered to be built for stability above all else-That will come with the downside that most of your software will be more than a few versions behind, except if it's something that is released for reasons of security/stability. Their software repositories are going to be among the largest, you'll likely find what you're looking for. If you're looking for something a little more recent, Debian testing ought to be fine for the most part, one isn't likely to be reducing stability on their machine. It's a little tougher for a novice to set up, but I do believe that resources are out there to assist in setting up a new install.
Ubuntu: Based off of Debian unstable. Many have been unimpressed with the last few releases, as a lot of the things that were working fine with the older version they had been using, breaks or doesn't work quite as well as it used to. They may simply be getting too large, they might have more people than say, a distro like sidux...But many may not be as experienced in linux as others might...And it tends to show when bug reports come in, and either get fixed half-assed, if they're addressed at all. I ended up having to fight with a fresh install far too often, and many things still didn't work properly afterwards.
Ones that I've settled on more recently have been SimplyMepis, (based off of Debian stable, works better on laptops that gave me fits with Ubuntu. Community generally is older, and gets questions answered without issue for the most part) and a lesser-known variant called Pardus. (Independent distro from Turkey, no 64bit..yet, but everything also worked with a laptop that has AMD/Geforce 9100 graphics...Works and looks very nice. Reportedly works nicer with Intel x3100/4500 integrated graphics as well)
Mepis might work better if you're looking to play games, I haven't really tried playing anything under Pardus as of yet.
These days, I can't really find myself recommending Ubuntu for the most part, even for beginners. For a grandma/grandpa box where they're going to be using email/web browsing, something like Debian or Mepis would probably work better. Mandriva is also another one to consider.