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The "app" version is a simplified way of using Sage. To use it you just click on the application, just like most others on a Mac, and it will start a Sage instance in the background, open a browser window, and launch the Sage Notebook.

The "non-app" version is a directory containing the Sage binaries and necessary libraries. The main binary opens an instance of Terminal and runs Sage in the terminal. It's less user-friendly but it allows you to examine all of the components of Sage and (possibly) develop for Sage as well. (I say "possibly" because I've only developed for Sage via a source installation.)

Now, the "app" version also contains the usual Sage directories, binaries, etc. but they're just slightly harder to find. So basically, the "app" version is the same thing as the "non-app" version except it's more user-friendly.