I had exactly the same issue with Linux Mint 13 "Maya" (Mate 32-bit edition).
Apparently, my web browser must already be open in order for the links in the Mint Welcome Screen to work (at least in my installation). I tried reinstalling mintwelcome; and tried setting my default browser application to "Firefox" (in Preferred Applications) -- but those measures didn't work. But I'm able to use the links in the Welcome Screen with the following procedure: After starting Linux, I close the Welcome Screen, then open my browser, then open the Welcome Screen again (in Preferences). I can then click on a Welcome Screen link such as "New Features", and then close the Welcome Screen after the relevant web page comes up.
Thank you Linux Mint developers for all you have done to make Linux Mint so terrific. I just coverted all three of my computers from Windows to Linux -- and what a difference!