Difference between Pop-Punk & Emo

I was having a bit of a dispute with two of my friends tonight, about the difference between Pop-Punk and Emo music. I personally consider Pop-Punk and Emo to be two distinct sub-genres of rock, while my friends believe that they are the same genre and bands such as Fall Out Boy, The Offspring and Green Day all belong under one genre. Their argument was that the lyrical content of those bands are the same and that even Linkin Park could be considered somewhat Emo (which I strongly disagree with). However, I believe that pop-punk and Emo are two distinct movements. The Emo scene has it's origins in Pop-Punk, however I don't think they are the same genre of music.

What are your thoughts on this? How would you define emo and pop-punk?