There is a 5 minute video from Sitepoint from Russ Weakley that is worth watching.
Basically REM uses the size from the root, where EM uses the size from the parent. Em can have compounding of sizes.
Note, this doesn’t work in IE 6-8.
Using PX first, then REM is a fallback for old IE (IE will only read the first value and ignore the second).
There's also a poly-fill library, but the fallback seems better.
Here’s the example from the video.
From the comments on the page “the point to take home is em/rem units are mobile friendly and accessible.” so using em/rem should help with a responsive design.