So I did try setting the "Allow Applications Downloaded Anywhere" option in System Preferences -> Security and Privacy previously and that did not work either. However i did get it working with the following procedure:
Procedure that works
1. Delete Mathematica.app from the Applications folder
2. In a terminal shell I executed the following command: sudo spctl --disable /Applications/Mathematica.app per instructions at https://discussions.apple.com/thread/4680288?start=0&tstart=0
3. Re-Install Mathematica 10.4
4. Set Security and Privacy to "Allow Applications from Anywhere"
5. Launch Mathematica by double clicking Mathematica.app in the Applications folder
6. Enter Activation Key
7. Mathematica 10.4 now working on MacOS X 10.11.4
It appears there was a problem in gatekeeper after I tried to launch Mathematica 10.4 the first time (perhaps I clicked cancel instead of ok when the dialog panel popped up saying Mathematica.app is from the Internet do you want to open?) In any case, it is now running. Thanks for all the comments and suggestions.