There are a number of old songs whose verses have been forgotten while the chorus lives on. Right now, I can think of these:

The Band played on
Bicycle Built for Two
When Irish Eyes are Smiling.

The verses tend to be 'talky,' while the choruses lilt, and people just seem to like the choruses better. It might be an interesting project for a neurologist or musicologist to look into this further.

I like to sing 'Let the Rest of the World Go By' once in a while, so thanks for posting.