Windowed game?

There's usually differences in the game or online code that prevents playing across the two platforms.

hello!

If the problem is that the digitally purchased game is telling you "Do you own this game or app" or please sign into the account that owns the game to play it. I would recommend you contact Xbox Customer support at
Xbox Customer Support so they can check the licenses on your account or help you find the account that was signed in while the game was purchased.

If the game is on disc please try these steps

How to troubleshoot problems playing a disc on your Xbox One

Hope these steps help please reply with an update if you still have issues

Thanks for reaching out!!

The same rules governing Live on the 360 govern those on GFWL. So providing you aren't cheating, glitching or otherwise doing something you shouldn't, then you wont get banned.

Only games that have been designed from the beginning to be cross-platform can be played on Games For Windows Live and Xbox Live at the same time. Only about one or two games have ever been cross platform in the whole of the Xbox LIVE/GFWL lifespan.

Another good reason would be that on Games for Windows Live you do not require a Gold subscription to play online, whereas on the 360 you do.