How different are the apps? Free tools like ArcGIS Online Assisstant can help you update things. There are built-in options for doing a find/replace on service URLs, but if you just want to completely replace one app with another, you can copy/paste the web app's JSON from one to the other. This would probably do what you want, but it's also possible you'll break something.
1. Select View an Item's JSON
2. Find the existing app (the one you're going to replace) and select it.
3. To the right are two sections: Description and Data. Copy the Data section and save to a local file. If you break something, you'll be able to use the copy to restore the original.
4. Now find the new app you want to swap in. Copy its Data JSON.
5. Go back to the app you're swapping out. Click the pencil, select all, delete, paste.
6. Make sure the app looks like you expect.
7. Restore the backup JSON if things aren't working.
That should be it! Your existing app at its same URL will now have the JSON of the new app!
Oh wow, I somehow never knew about this... Time to break some apps while trying to fix some insignificant details...