Cannot launch Steam game with Wine
The difference may be related to whether you're using the Mac driver. Go to Set Screen Options, and check Use Mac Driver instead of X11. It's the top checkbox under "Other Options". 

Otherwise, it's not clear how your installations are different. Because you're using Wineskin, however, you have a great alternative: Just copy the Wineskin app from your working computer to this one. That will carry along all of the settings and tweaks you have in your working copy. It's all self-contained!