h265 (HEVC) is a more efficient video compression standard than h264 (AVC).
So the advantage is its better at what its designed to do. A video using h265 can have higher quality with similar storage size or similar video quality using less storage space....OR anywhere in between when compared to h264.
I grabbed a random video out of my photo library and ran it through Handbrake using both h264 and h265. I used all the default settings for both so its a MKV container using the avc1 and hevc compression standards.

(click image)
With no noticeable quality difference the h265 is using ~27% less storage space.
The disadvantage is compatibility. The above HEVC plays smooth as silk with VLC on my iMac which does not have hardware support for HEVC.
However the same file played using VLC has too complex of an encoding for the AppleTV4 to decode. This is a screenshot through quicktime of my AppleTV4 output signal attempting to
AppleTV4
iPhone 6S (no true hardware support for HEVC) plays the HEVC video file fine in VLC.
iPhone 6S
On the AppleTV4 screen grab aside from the obvious compression blocks its playing at 5 fps with long pauses. This is especially bad since that is using the main profile, low resolution and only 30fps.
The real disadvantage about that is consistency. I'm assuming VLC isn't optimized very well for tvOS 13. Or there is some other limitation there. I question the optimization of HandBrakes h265 implementation too. Relying on 3rd app devs to optimize their apps so your videos play isn't a good situation to be in.
Another disadvantage is the time it takes the transcode the video using h265. Seems to take my MacBook about 2x as long with hardware support but it takes my iMac 5x as long without hardware support.
My opinion on the matter, we've been working with h264 and 1080p for so long that we've made appropriate accommodation for its file size. Now with 4k HDR we need something more efficient and that is where h265 comes in...
EDIT: The AppleTV 4 playback problem is VLC. I can see it dropping a couple frames but Infuse can playback that 1080p HEVC at a level that I would consider "watchable".