The simple answer is tyre width matters quite a lot and it is why the pro peloton has gravitated towards 25mm or even 28mm tyres when 23mm or less always used to be the standard. There are three main factors that make wider potentially faster: rolling resistance, comfort and aerodynamics.
When looking at rolling resistance, the key factor is tyre deflection. Any tyre is flattened under load, creating a flat contact area, which is detrimental to tyre rotation as it causes the wheel to lose ‘roundness’ and increases rolling resistance. If you pump a 23mm and 28mm tyre to equal pressure, the contact area will be the same, but the wider tyre is flattened more across its width, whereas the narrower one has a longer contact area, and greater resistance.
So, a wider tyre allows you to run lower pressures with no rolling resistance penalty, allowing for a plusher ride. Although not directly upping your speed, it’s easy to see how having to contend with less road buzz could be significant, especially over a long bike leg.
Finally, with modern wider rims, a wider tyre creates a smoother transition between tyre and rim. This can be significant aerodynamically, and faster.
How do I choose the right tyre pressure for different races?
How to repair a puncture quickly during a race