Well, since you can better modulate the power being applied to the disk, you will stop sooner. Not by a huge difference but a disk brake gets the job done a little sooner.
That's why I said disk brakes are better then rim brakes. Traditional cantis are not bad, but disks are better.
Rims *are* discs.
Modulation isn't a function of where the disc is located. It's almost entirely a function of how force gets transmitted from your brake levers to the pads, not whether those pads are acting on a 180mm disc or a 700mm disc. You can buy brakes operated mechanically (cable-actuated), or hydraulically in either format. In fact, if you really, really, were interested in improving the modulation of a common mechanically-connected rim brake, you needn't go to road hydraulic rim brakes; you could replace the brake cables with low-friction cables with compressionless housing, like discussed
here.
That's modulation. If the tires are the same, the other issue is pad composition and disc rotor surface. Just as there are alloy and carbon wheel rims, there are alloy and carbon disc rotors. Carbon in either format has challenges as a braking surface -- they tend not to dissipate heat as well as alloy. The GCN video compared a hydraulic actuated alloy disc brake with a mechanical actuated carbon rim brake, and found no difference in dry weather -- but differences in wet weather. If you were concerned about braking in wet weather (and even though I live in a place that has been experiencing a multi-year-long drought, I'm in this category) the real lesson is that braking on carbon surfaces with current technology isn't ideal. That's true whether the braking surface has a 180mm diameter or a 700mm diameter.