A lot of marketing involves hype around lighter parts, and you can spend almost infinite £ ($, €, whatever) in trimming a few grams here and there from the bicycle weight. However, the heaviest part of a bicycle by far is the rider. The weight and physical fitness of the cyclist plays a much greater role in performance than the weight of the bicycle. I've passed some rather unfit people uphill riding expensive-looking road bikes while on my heavy 3-speed laden down with books.

Some of the perception of a lighter bike being faster and more agile might come more from the frame geometry. A rather heavy city bike will have more relaxed frame angles than a racing bike, which has more aggressive geometry and also happens to be lighter.

For competitive cycling, however, the cyclist is already at the peak of physical fitness, so these tiny weight reductions from lighter frames, 'aero' wheels, unobtanium bearings etc matter a lot more.