I read it's twice as efficient as AVC
Apple has been saying it saves about 40% space, but other stuff I've read lists 30%-50%, although usually closer to the 30% than the 50%.
However, it should be noted that Apple's numbers refer to 1080p video. I've read that most people don't try to compare against h.264 for 4K, but h.264 was never a good choice for 4K in the first place.
BTW Red Squirrel, it should be noted that Apple is switching to the HEIF image format for photos as of fall of this year, for any recent iPhone running iOS 11. HEIF uses HEVC encoding to create the images. So as of this fall, there will be a bazillion iPhones spitting HEIF images (as .heif files). This is gonna wreak havoc on image sharing sites methinks.

I betcha they're all scrambling now to get HEIF image compatibility in place before iOS 11 hits the airwaves. Why switch to HEIF? Because it uses about half the space as JPEG for the same quality.
I've talked about Apple a lot here, but it should also be noted that the Samsung S8 Android phone also supports HEVC.
So, while HEVC isn't exactly an old format, there is going to be a LOT of HEVC-based media out there in very short order. And as mentioned, Netflix 4K and 4K UHD Blu-ray is already HEVC. In addition, Amazon Prime uses HEVC not just for 4K, but for their other resolutions as well.
It's for this reason I did not buy into Broadwell or Skylake for new computers. Broadwell is essentially crippled when it comes to hardware HEVC support, and Skylake has a bit of a lame limb. Full support didn't come until Kaby Lake, which is why I've waited so long to buy new machines. Last week, I ordered both a Kaby Lake Core m3 7Y32 laptop, and a Kaby Lake Core i7 7700K desktop. See below for an example as to why: