It depends on the specific weather conditions and how it affects the field.
I'll start with the fields. Newer stadiums use field turf and footing is only slightly different. There might be the occasional slip, but for the most part this won't affect anything. On real grass the field can get muddy and slick. Because the DBs need to react they have a tendency to slip until they wear longer cleats. This keeps them from slipping but they also lose speed. Faster WRs who run good routes will benefit. Snow will change any field to slippery conditions.
As for the weather conditions, light to moderate rain with little wind doesn't really affect too much. But heavy rain pushes the ball down and calls for shorter passes. This means flankers who run deep routes will not get thrown to as much. Snow also can affect distance because of the temperature drop (I can't remember how this works, but I've read it before). And obviously in strong wind conditions you're not going to throw the ball deep unless you really need to or your guy is wide open.