AS6706T have iGPU, if you need to reencode before stream, this is the only choice between these 2 NAS.
Intel Celeron N5105 seems slightly faster than AMD Ryzen R1600, but if just for file transfer then likely the difference is not noticeable.

DS1522+ have 1 more 3.5 bay while  AS6706T have 2 more M.2 SSD and I believe it's not locked to any specific M.2 SSD unlike Synology need to use it's proprietary SSD if for storage. The LAN port is also 2.5GbE and there's PCIE slot that's not proprietary so can use 10GbE card for cheaper if you need the speed. DS1522+ also PCIE slot for 10GbE card but it's proprietary.

OS and software wise undoubtedly Synology is more polished and have more apps that can give similar feel to most of Google apps. If you plan to use the NAS apps than Synology have the upper hand, if you going for open source apps and run everything on Docker then both can do it.

You can try Synology OS demo and also it's apps demo by using this link below
https://demo.synology.com/en-global/dsm

Asustor also have it's live demo site for you to try the OS and it's apps
https://www.asustor.com/live_demo

just to add on xxboxx's post, on the streaming part, depending on your configuration and playback device, you might not need to be concerned about reencode

my plex setup, playback devices consists of the following:
4k smart tvs
windows pc using plex app (using browser sometime requires reencode)
ipad (bought the app)
mi box s

plex relay disabled, external playback through vpn, all clients are considered local
so far all video streams are direct play