Both of those Linux distributions are based on Debian/Ubuntu Linux so there isn't a big difference under the hood.
Although, if I were you, I'd go with Mint for two reasons. Mint is currently the most popular Linux distro (see
distrowatch) and I think you'll find the support community for Mint is a good one. The 2nd reason to go with Mint is that the default Desktop is Cinnamon, which is basically Gnome shell with all of the more popular extensions already installed. Given that, I believe Cinnamon will make the transition from Windows a little easier than jumping right into a distro with plain Gnome3 as the default desktop. MATE desktop is also available with Mint, which is another popular traditional Windows style desktop. Mint also includes most of the media codecs you need to get you playing music and videos right out of the box.
As YAFU said though, all it takes is a CD to give any of them a test drive.
Good luck and welcome to Linux!