IME, some bikes "feel" faster than others, but usually at the end of a ride I am not able to measure a significant difference.
However, I don't race. In a race the only real "measure" is your placing, so all things being equal (including fit) a lighter bike is an advantage in most cases.
Bike weight does make a measurable difference when you're accelerating: i.e. sprinting/accelerating out of a corner/climbing/etc. So if your rides involve a lot of climbing, corners, or sprints then a lighter bike will probably make a difference.
(Note: it's true that you're constantly "accelerating" even when you're riding at a constant rate of speed in the sense that you are counteracting the effects of air resistance and friction, but if that type of "acceleration" is affected by bike weight at all, the effect is small.)
In many cases, aerodynamic drag makes a much bigger difference than the weight of the bike. Focusing on being able to maintain a position on your bike to minimize drag, while maintaining your power output, will likely make a bigger difference than buying a lighter bike.
(Unless maybe you're doing climbing TTs at a speed that's low enough that the effect of lower weight is greater than the effect of lower drag.)