Absolutely, you can do both. WeatherAds enables you to play/ pause ads based on the weather, as well as automate bid adjustments based on weather as well.
However, it’s worth noting that For AdWords, WeatherAds works at the campaign level, and for Facebook, the weather targeting works at the ad set level.
This means that for AdWords you would create separate campaigns for each of your weather based ad variants (e.g. campaign 1 would contain the ‘sunny ad’ and campaign 2 would have the ‘rainy ad’.)
Then you would import both of your campaigns into WeatherAds and apply rules that state 'only play campaign 1 when it is sunny' and 'only play campaign 2 when it is rainy'.
For an AdWords campaign, once you have selected your condition, you would simply leave the bid modifier at 0% and then select 'don't show ads' If conditions across filters are not met.
For Facebook the process is exactly the same, except in place of 'campaign', read 'ad set'. With Facebook weather targeting there is no bid modification, so you would simply set up your weather targeting rules for each ad set and then select either 'show ads' or 'don't show ads' when conditions are met.