The default install of Xubuntu is probably a bit lighter weight than Mint. But that's because XFCE is lighter than Cinnamon, XFCE is the default desktop environment of Xubuntu.
You can install XFCE on Mint and use that instead of Cinnamon, so it's not as if you're stuck with your choice.
So why not
take a look at Cinnamon and if you like the look of it go with Mint. If it's too slow install xfce, reboot and select that at the login screen.
For a really lightweight install, crunchbang and archbang are excellent choices using Openbox. I've used both. They're highly configurable but the default installs are completely fine. These are, respectively, Debian and Arch with some configuration to make them more immediately suitable for general desktop use, as vanilla Debian and Arch installs usually need a bit of tweaking to get them just right. "Just right" is subjective! They're just right for me though.