It makes a big difference, as far as aerodynamics are concerned; wider wheel will result in a less bulbous/light bulb shape, which will help the aerodynamics. It will also help turn-in and general confidence.
I recently went from 15mm internal width rims to 17mm internal rim widths, both with 25mm tires (so even with the new wheels, still pretty light bulb shaped), and I still noticed an appreciable difference in turn-in response when leaning into a corner. Then again, I also swapped tires and tubes at the same time, and am on much lighter/stiffer wheels, so it's hard to quantify what parts are primarily responsible for various changes in feeling/performance.
That said, I don't really know what is truly considered the gold standard now for widths; it seemed like for a short while C17 was all the rage, but it seems we are already moving past that into even wider rims? Someone else can chime in on that. I imagine a C17 with a 28mm tire wouldn't be any better than an old C15 rim with a 25mm tire, so I suppose as 28s become more the norm, so too will C19s or whatever.