Covers that are arguably better than the originals:

"I'll Be Seeing You"

Moses Sumney does a fabulous performance of this classic.

Liberace played and sang "I'll Be Seeing You", at the end of his TV series which he made his theme song.

Francoise Hardy & Iggy Pop also do a wonderful recording with a vintage geranium diode fuzz box guitar part.

 
Hey Jude by Wilson Pickett. I prefer his soulful rendition to that of the Beatles.
As a bonus, there's brilliant lead guitar by Duane Allman at the end. I heard a story that Clapton listened to the record and exclaimed "who is that playing?" "I want to meet him". Thus was born Derek and the Dominos...