[user104658]

Sounds to me like the router is trying to assign the same local IP address to both devices (technical term is IP Address Conflict). When both are on, it gives the PS4 a different address but then when you switch off the TV, the router defaults his connection back to what it "wanted" to be and that would briefly disconnect him. It's probably happening to both of you. It would be a split second which wouldn't interrupt TV streaming (because of buffering) BUT it would reset any connections which would mean getting booted from an online gaming session... So it's likely happening both ways, just with the smart TV it's so brief you don't notice.

The solution is to log onto the router from a computer when both devices are ON, find them in the list of connected devices, and assign at least one of them a static IP address out of the default range. I'd just set both. 192.168.0.101 to 192.168.0.199 are a good bet for avoiding clashes.