Chrome’s auto-update system is for the most part unobtrusive — you see the three-bar symbol in the top-right corner go from green, to orange and finally red, until you restart the browser. So, you cycle the program and Chrome fails to fire up again and when you do it manually, nothing has changed. The problem is easy enough to fix however, without rebooting.
If possible, you should let a program exit naturally, especially those that save data to the hard drive — word processors, image editors, etc. Chrome, on the other hand, doesn’t really need to store anything that important, so terminating via Task Manager is usually OK.
In the case of Chrome refusing to update after a restart, it’s almost always due to the program not shutting down in the first place.
Load up Task Manager and look for instances of “chrome.exe”. If you spot any, try to find the one using the most memory as it’ll almost always be the main Chrome process. If you kill that, it’ll shut down all the browsers other instances, allowing you to start it up again and for the update to complete.
And if that fails, well, you can always nuke it from orbit (read: shut down Windows).