If your connection to the worldwide web comes from a router situated safely inside your home, you're generally not affected by any type of weather. Usually, the climate inside the home is conditioned to be optimal. Only in cases of extreme heat or humidity will your connection be severed. Humidity can affect the speed with which the signal travels through your house, while extreme heat (again, unlikely even with central heating) can cause the router itself to overheat.
The thing that affects your router on bad weather days is most likely traffic. Just like yourself, everyone else is also surfing the web instead of the ocean. An online connection isn't reserved for the phone and the computer anymore. Watching TV and listening to ambient music all day long also happens online, and that takes a toll on internet servers.
If there's public WiFi in open areas in your city, it may be affected specifically by rain. According to Techwalla, raindrops absorb the 2.4-GHz frequency, upon which many wireless setups rely. The closer you are to the router, the steadier your signal will be.
If your internet is cable- or satellite-based, it is easily affected by the bad weather. Satellite signals can't penetrate dense, solid objects, that includes trees or heavy rain.
In the case of cables, they can simply break during a prolonged freeze. Surprisingly enough, fog, snow, and hail won't have a significant effect on your internet, as they aren't as dense as rain. The water doesn't absorb the signal, but rather breaks and scatters it.