I implemented an app that uses Web App Manifest and Service Workers and uses material design lite. It's almost indistinguishable from a native Android app. If I put a little more effort into the animations it would be even closer.

The biggest gap I find is sounds. There's no way to say "do the native click sound here". Now I could find the sound and play it in an <audio> element but I'm sure that these sounds differ depending on manufacturer and surely differ on iOS and WinPhone.