I did a little research for you on the internet, and I have a theory to the problem.
Your Galaxy Tab 10.1 is a very old device,
And it is probably operating on the 4.04 software!
That new version of You Tube may be to advanced for the software in your tablet!
That's why play store is telling you it is installed but it still will not work!
My best suggestion is to...
First go to play store, turn off your auto update setting. You will find it in the menue for that app.
Then go into settings, then apps manager.
Find You Tube. Then uninstall all updates.
That will take You Tube back to a version that is compatible for the software in your tablet!
If I'm correct, You Tube should work just fine.
Now that auto update setting I told you to turn off! Leave it off! Or you will get more updates that will cause similar problems.
Let me know if this works!