The game itself does not determine when to change the weather. What you do during the game determines whether the game itself should change the weather.
For example by sitting on benches you can change the weather or by sleeping in your bed at Hotel AZ. My source for that is here.
For when it is sunny Pokémon like Helioptile, Eevee, and Skarmory will appear during this weather condition.
Next is Cloudy. No Pokémon have a higher chance of spawning in like during the sunny weather. However, the weather being cloudy in the game doesn't necessarily mean that it's going to rain soon. If you want to read more about it you can go to my source here.
Rain is up next after cloudy. Apparently during the rain water type moves are boosted by 50% and fire type moves are reduced by 50%. My source for that is here and then of course you can evolve Sliggo into Goodra during the rain if it is above or at level 50.
Fog is after rain in the list and nothing new or different changes during this whether unlike Rain and Sunny whether.
Next up is the Rainbow weather after Fog in the list. There are no new different conditions or Pokémon appearing in this weather condition.
Hope this helped!