The weather will influence what spawns around you. If a Pokémon spawns because of weather patterns, it’ll have a more intense ring on the map and an icon next to its name during capture (see red arrows).
– Clear and sunny: More grass, ground and fire Pokémon will spawn
– Cloudy: More fairy, fighting and poison Pokémon will spawn
– Foggy: More dark and ghost Pokémon will spawn
– Rainy: More water, electric and bug Pokémon will spawn
– Snowy: More ice and steel Pokémon will spawn
– Windy: More dragon, flying and psychic Pokémon will spawn
Niantic tells me the weather influences spawns, but it’s not set in stone. You might see snow Pokémon pop up when it’s sunny… they’ll just be considerably less common. Meanwhile, weather will impact spawns of all Pokémon, not just the new Gen III stuff.