Does a stove top fan really help to cool the stove?

The purpose of a stove top fan, or vent hood, is to pull smoke and odors away from the cooktop, not to cool the stove. Ideally, the fan vents to the exterior of the building in some way to make the kitchen more pleasant.

If a stove top fan does not vent to the outside, it simply scatters the smoke and odors around the kitchen rather than taking them away from the cooking environment. Turning on the fan in the vent hood also creates a fairly loud noise in the kitchen, so fans that do not vent to the outside are basically useless in the kitchen.