This study took a hypothetical rider and had the them ride two bikes up a hill. The first bike weighs 15 pounds and the second bike weighs 11.79 pounds. Each test the rider will ride at 15 mph. Everything is kept constant except for the bike. What we see in the first image is the reduction of power required to get up the hill. There is no noticeable difference between power output until the rider reaches a 10% gradient, where the difference is noticeable but minimal.
There is reasoning behind this. If you want to reduce required power by 2%, you have to reduce total mass moving up the hill by 2%. For a 150 lb cyclist and a 15 lb bike, 2% of mass is about 3.21 pounds (this happens to be the difference between a 15 lb and 11.79 lb bike). 500 watts is the power requirement to maintain speed up a 10% gradient. Weight to power savings ratio is nearly linear, so we should expect a one-to-one relationship.
It is true that climbing up hills that might only be a degree or two of incline makes riding a bike feel much harder. So it is logical that people think having a lighter bike will help them conquer hills. But you'll only realize power output reduction at a hill with a 10% gradient or more, and even then you'll only see a percentage or two difference.
It's easy to discredit this information. I mean, the professionals all have the lightest and latest gear and it makes them fast, right? Those differences must matter!
Yes, differences matter for professionals who are only separated by a few seconds in total time after riding 2000 miles. A 1% or 2% differences is huge for them. But for the average rider, it really doesn't make a difference.
This information was adapted form the book FASTER by Jim Gourley.