An extraction fan takes the old air out, the old air is warm and humid and smelly.
If you use a filter then that cleans the smelly air, that's not needed for everyone but even in legal grows its not advised to advertise your growing and get yourself robbed.
Replacing the old air removes the heat and humidity, the old air is depleted in co2, a vital part of plant growth. New air replaces the old air, its higher in co2, has less moisture and is cooler.
Circulation fans help strengthen the plants branches and prevent microclimates around the leaf which would be warmer , this helps minimise Powdery mildew , mould ect, it also helps increase transpiration which helps nutrient uptake.
People are the same, we get hot in a no wind situation and we have a microclimate a few mm over our skin, that is warmer than the rest of the room, so we use a fan, the air is not cooler but it removes that microclimate which helps us cool down and aids perspiration which cools us down.
So yes, ideally you need both , you might get away without , you might manage your climate better without extraction but you raise the risks of losing a crop or not getting the most out of your equipment.