In a fall, roughly the same load is applied at every point along the rope, at the climber's harness, and at the anchor. "Roughly" means that this is an approximation where rope drag is negligible, the mass of the rope is negligible, and we're not taking into account the geometry of a redundant anchor. (If the anchor is equalized, the load could be shared by the different pieces, but if the angles aren't favorable, the load could be increased.)
The fall factor and the maximum load are not independent parameters. WP has a formula that expresses the maximum load in terms of the climber's body mass and the fall factor. Since people's body masses are all on the same order of magnitude, the fall factor almost completely determines the maximum load.
Physically, the maximum load is probably the most relevant thing that determines what bad things will happen to the rope, climber, and gear. (It's also possible that total energy dissipation could make a difference.) But since maximum load is essentially determined by fall factor, all you really have to worry about is a single number, which is the fall factor. This is good, because fall factors are easy to estimate while climbing, but it's not practical to estimate while you're climbing how big the dynamic load will be from a lead fall.