Whenever I try to like a video through the YouTube API, it flags the video as liked but the like counter doesn\'t increase whenever I do. It tells me that I\'ve liked the vi
Easy explanation: Youtube doesn´t count the likes live, as well as the views. They are periodical updated after a certain amount of time. This enables google to analyze the likes / views and filter suspicious actions. Remember the 301-views limit of the videos, a few weeks ago they changed it to display 301+ to indicate that there are more views already, but the count will incrase later on.
So if you wait a few minutes/hours your like should appear, but you cant be sure its really yours since anybody could have liked it too in the meantime ;)
Do you need a more detailed explanation?
Maybe watch this youtube video http://www.youtube.com/watch?v=oIkhgagvrjI to understand that principle.