The definitive answer: It depends

If the new road bike is lighter, fits you well and has gearing that you're comfortable with then your climbing, etc., should improve. If you buy a cheapo road bike that weighs more than your hybrid, doesn't fit and gearing that you're uncomfortable with - obviously - no improvement.

As a rule of thumb, however, a road bike will put you in a more aerodynamic position (leaning over) and that can improve speed. On the minus side, it can be uncomfortable if (1) it doesn't fit well or (2) you have issues with neck, hands, shoulders, etc.

A road bike should be lighter than your hybrid. That can increase performance.

You would probably want to go to a good local bike shop, tell them what you are looking for, your price range and test out many different bikes. One size does not fit all.