If you lose power the WS-2902A display console will turn off the LEDs but continue to communicate with the outdoor sensor array. It will keep track of rainfall and highs and lows. If you hit the button it will turn on the LEDs momentarily. BUT...it will not connect to WiFi, not because the WiFi is probably out too on a power outage but because it actually turns its WiFi off. So no Internet reporting without AC power and it doesn't keep internal logs (just has totals) so there is no data to send when power is restored. Which means your display will have highs and lows and rainfall but it will not match your Internet published data.
The WS-2000 does have internal memory and real full data logs. If you just lose Internet service but still have power then maybe the WS-2000 can send the backlog when Internet comes back.... MAYBE... I don't know.
Also just because the WS-2000 doesn't have battery doesn't mean you can't give it a battery by plugging it into a battery backup (UPS) like you would a desktop computer. Of course I then also recommend adding a UPS to your Internet modem and WiFi router. A UPS normally will only keep a desktop computer running for 10 minutes, enough to save work and safely shut down, but it will probably keep the WS-2000 display console running a very long time, hopefully long enough to endure a brief power outage.