"I have the ability to climb a tree" means I can climb the tree, I am able to, if I want to. Maybe I have learnt to climb, maybe I was born to climb. "There is a possibility I will climb the tree." Maybe I will, maybe I won't. It's possible.

Ability refers to your physical or mental capacity. If you say you have the 'ability' to climb a tree, it means that you are tall, strong, skillful or agile enough to do this. Possibility refers to how likely something is. We use it to say that something might happen. The weather forecasters would say 'It'll be sunny this morning but there's a possibility of rain this afternoon'.

I will use them in two sentences. Hopefully you will understand the difference I have the ability to climb that tree There is a possibility that I would climb that tree Got it?