Is there any benefit to having "NAT Type 1" (aka no NAT and publicly-routable IP) on a PlayStation?
Solution 1:
A conclusive answer... probably not known except to developers for the PS4. However...
NAT punch through is basically a solved problem now a days. You most like will never encounter a problem with NAT Type 2 versus NAT Type 1.
There'll be a negligible latency difference (benefit to Type 1, due to not having a piece of hardware/software extra) that you'll also never actually see.
So.. No, probably no benefit, maybe there's some game that isn't written very well that could use the help. If you find one, set it up as it's own IP.
However, there is a downside of giving something it's own public IP address. That is, it is accessible from the internet. With a NAT you can only be accessed if you first start a connection, without anyone can access you. Someone might find an exploit for PS4's you might become the victim of by hooking it directly to the internet.
Also, any LAN features a PS4 has won't be accessible if you put it directly on the internet instead of behind the same NAT as your other devices.
So, the benefits are negligible and the downsides are real. Put it behind the NAT with your other devices and don't worry about it.