Intuition is important. It is how we "do" mathematics. It is how we "feel" it and "see" it. It is our eyes, ears, and hands. What are proofs then? Proofs are how we build mathematics.

Intuition will basically always give a reasonable mathematical result. However, it does not accumulate. If you build everything with intuition, you quickly wind up in contradictory circles. You can fix these contradictions, but you'll end up spending most of your time doing so, since every time someone has a new intuition, you have to check it with all the other results.

For example, take the Banach-Tarski paradox. Intuition would have quickly eliminated it as a possibility. Then when Banach and Tarski came along, any chunk of mathematics based on the intuition that it was impossible would need to be rewritten. It could be done, but I hope its clear that this would be infeasible to do regularly.

Of course, this would probably also happen with Fractals, probability (many times), set theory (more than necessary), topology, computer science (many times), mathematical logic, etc...

How do proofs solve this? Because we know if a proof is wrong very quickly (usually). We do not have to wait to see if someone has a different intuition/proof, we can just check the proof. Sometimes proofs can go a while without being corrected, but they are usually repairable. Therefore, when using proofs, we can fairly safely build up mathematics without worrying too much about lasting errors. In fact, if we agree on some formal system (like ZFC), it becomes even easier. Of course, we are not writing completely formal proofs (yet), but proofs are less likely to conflict if everyone has the same theory in the back of their head. Then the only results that are only decidable with intuition are the axioms, which is usually a small list of statements.