Apple - AppleScript app started prompting to run or quit. How can I disable this?

When you create a script in AppleScript Editor and then save it as an Application, there is an option to show a Startup Screen, as shown in the image below:

When that option is enabled, when you launch the script in the Finder, it will first show that initial dialog before continuing.

As long as the AppleScript script-application in question wasn't saved with the Run Only option (see the image above), you should be able to open the script in AppleScript Editor by dragging the script-application's icon onto AppleScript Editor's icon. (When a script is saved as Run Only, AppleScript Editor won't allow you to open it to see the source code). So, if you are able to open it OK, then you could Save a copy of it and make sure the Startup Screen option is disabled.

As for why this suddenly started happening when yesterday it didn't, I'm not sure. What version of OS X are you using?