Solar panels' electricity production varies depending on the amount of sun hitting them. Cloudy days should still produce power, but at a reduced amount.
Snow coverage will usually block production, but panels heat up and shed snow much faster than metal or shingle roofs. So depending on the amount and temperature, most people are back to producing within a couple days of a snow storm.
Here's a really cool, unaffiliated video showing solar panels in the snow - https://www.youtube.com/watch?v=UzhJvcNdwM8