Conflict is a struggle or clash between opposing forces. It can be internal (within a character) or external (between a character and an outside force). Conflict is what drives the action of the story and creates suspense.
Climax is the turning point of the story. It is the moment when the conflict is resolved, or at least comes to a head. The climax is often the most intense moment of the story and leads to the falling action and resolution.
In short, conflict is the engine that drives the story forward, while climax is the peak moment of the story.