When you create a Mac application project in Xcode, Xcode sets the deployment target to the version of OS X you're running. The deployment target is the earliest version of OS X that can run the application. When you created your project in Snow Leopard, the deployment target was set to 10.6, which means the application will run on 10.6 and later. When you created your project in Xcode 5.1, the deployment target was set to 10.9, which means the application won't run on earlier versions of OS X. That's why your project runs when you create it in Snow Leopard but not on Xcode 5.1 on Mavericks.
To support earlier versions of OS X, the first thing you must do is set the deployment target to the earliest version of OS X you want to support. If you set the deployment target to 10.7, the project will run on 10.7 and later, assuming you don't use any technologies or function calls introduced in later versions of OS X. For example, if you create a SpriteKit project, it won't run on anything before 10.9 because Apple introduced SpriteKit in 10.9. Xcode will not tell you when you use function calls introduced in later versions of OS X. You are responsible for making sure your code uses only functions available in 10.7.
You should also check the deployment target for your xib files.
To support 10.7, you must turn off base internationalization for your project. Base internationalization works only on 10.8 and later. To support 10.6, you must turn off auto layout for your xib files.
You can access your project settings by selecting the project from the project navigator on the left side of the project window. You can access xib file information by selecting the xib file from the project navigator and opening the file inspector. You can find more detailed information in the following article:
Supporting Earlier Versions of OS X and iOS