I know you all probably think I'm way off, but this is how I personally relate to the song.
There is a guy and a girl (forget the video for now okay?) and one is in love with the other, but the other is afraid to admit that their in love as well. One of them doesn't use their heart as much as their head, and the other only uses their heart. In the end. It doesn't work out. Because The Scientist doesn't take a chance on love. So when it's ended, they both hurt, because they were best friends, and in the end. The stubborn idiot realizes he was in love all along. : sigh: ahah. I know. Probably way off base, but that's my rendition :)