OK here is what I found. By default Github pages do not publish files with certain file name structures. I found this document to explain and give a work around.
I don't really know what jekyll is, but I put in a .nojekyll file (the .nojekyll file is blank) in the root of the repository. I added it to the gh-pages branch too but for some reason it did not take. I deleted the gh-pages branch and recreated it and it looks to be working now.
Chrome Developer Tools are showing this:
1. The one that works is getting init.js from http://js.arcgis.com/3.12/init.js
2. The one that isn't working is trying to get init.js from http://timw1984.github.io/http://timw1984.github.io//jsapi/jsapi/init.js and showing a 404 file not found error.
I fixed that but now other errors pop-up. I guess building it on my own machine, makes it look for all the resources locally instead of the ESRI hosted resources.