That is a very good question and what's good for the good is not good for the gander. Cars being goose and bikes being gander, that is.
Lots is known about tyre width, particularly wrt racing cars. Racer cars like F1 have wide tyres so that they can get reasonable life out of them and also so that the heat generated during acceleration, cornering and braking can be extracted/;spread over a larger surface. The same amount of heat is generated whether the tyre is wide or narrow, but if there is more tyre, there is better air cooling and less sudden heat build-up during say a session of hard braking.
Yet, width does affect bicycle tyres too. Firstly, a wider tyre has less rolling resistance than a narrow one. The biggest reason, I think, is because the rubber deforms less and has less energy losses (hysterisis) when the tyre is wider. But don't think you can go on making it wider and wider. The sweet spot is probably around 28mm if you stick to a thin, non-armoured, non-armadillo type tyre. Puncture protection just adds thickness which adds hystererisis.
The second reason I think a wider tyre is slightly better is because it may grip better under certain conditions. Note that I'm not saying that there is a difference in the total amount of friction available in a narrow or wider tyre of the same compound and structure. Friction - therefore cornering ability or stopping distance remains exactly the same for wide and narrow. But, if you have a contact patch of 200mm^2 on your 28mm (wide) tyre and a contact patch of 150mm^2 on your 20mm wide tyre and you corner over a tiny slick of oil of 100mm^2, you will have 100mm^2 in reserve on the large tyre and just 50mm^2 on the narrow tyre. Should the 50mm^2 thus just not be enough to keep traction, you will slide on the narrow tyre but not the wider tyre.
But have a careful look at the scenario. It is pretty unlikely and probably far-fetched. Also note that the relationship between tyre width and contact patch size is not linear with the increase in tyre width. A 40mm tyre doesn't have a contact patch twice the size of a 20mm tyre. The increase is quite a bit less than doubling.
You may or may not be better off sticking to your Conti's. The wider tyre will NOT give you better traction unless the road is softer than the tyre. That's not the case in city commuting. Wider tyres give you benefits such as pinch-protection and comfort, which narrow ones don't.