After a recent windows update, the TeamViewer application seemed to believe I was using a new computer. When I attempted to login through the "New Interface", it asked me to check my email and add this device to my trusted devices list. After doing so, whenever I attempted to login, the login screen would spin and kick me back out to the front of the app.
After I turned off the "New Interface", the reason I couldn't login became apparent. It seems the application wanted to display some kind of popup to tell me that my device limited had been reached, (Because it thought my computer was a new device). That popup let me open the interface, remove the duplicate computer, and login.
So there may be a bug in the new interface where that popup does not display and you end up frustratingly attempting to login without any hint as to why it was failing.
I hope this helps you find/resolve any issue!