Well, in our case, it was a lot easier to buy Ipods for our kids rather than a less expensive mp3 player because all of our CDs are already ripped to iTunes, and in the iTunes format. Getting the music onto a different brand of MP3 player would involve converting the music to a different format or re-ripping them to our computer again. Lots of extra time and effort and space on our hard drive = lots of headaches.
Our kids don't download music with iTunes cards because they don't have the iTunes passwords

Because they don't have the passwords and we download the music (and apps and videos) to their iPods, we control what they get to listen to (which means mostly classic rock, 80s and showtunes, not too much current rapcrap).
There are lots of useful apps and fun games, and I don't think that most of the $50 mp3 players offer that (though now with Android, maybe that's changed?).
It doesn't have much to do with them having what everyone else has...it has to do with them having the technology that Mom and Dad are used to and that is easier for Mom and Dad to deal with.
And BTW, when they were that young and got their first IPod, it was actually an older one that we were no longer using when we upgraded ours!