It's a matter of degree.

My 32 lb touring bike felt a lot lighter AND much easier to climb hills with than my 32 lb touring bike loaded down with another 35 lbs of gear. And my first road bike, at 24 lbs, felt noticeably lighter and easier to climb with than the touring bike. When I first built up my current road bike, it was about 18.5 lbs. It definitely felt a lot lighter, but climbing felt pretty much the same. It's easier to fling the bike around when out of the saddle, which is what's really noticeable about the weight of a bike when actually climbing.

I think the major difference in feel between all of these bikes has more to do with geometry and fit than weight. A touring bike will be biased toward comfort; a road bike toward power. Other bikes will generally fall in a range between these extremes, and the effect of weight depends upon your goals. So, assume the same geometry, for the sake of argument: if you're touring or just riding around, then a lighter bike is noticeably easier to climb with, because you are typically seeking ways to reduce your effort without just going really slow. You're not going to going flat-out, because you want to enjoy the scenery, or because you don't want to wear yourself out completely from riding all day every day. Pushing more weight up hill after hill, day after day, is noticeably tougher than pushing less weight.

If you're racing or otherwise looking to get the best possible climbing performance, then the point about effort is academic. There is no difference in effort, because you're working as hard as possible, period. The difference under these circumstances is speed. You're going to climb faster at max effort on a lighter bike, period. When I'm going hard on my nice bike, it doesn't feel different than my old heavy bike. I still want to puke. I just get to the top a bit faster.

Keep in mind that these differences are a matter of perspective, not physics. The physics are the same. Same effort + less weight = more speed. Same speed + less weight = less effort. Same terms, just arranged differently.