Conjecture is knowing something with little or no facts to support your idea. Intuition is knowing something by using feelings and emotions instead of facts or logic. You use your heart instead of your brain. Conjecture is usually used in a negative way. Because your idea is not supported by fact, people will not believe in your idea. Intuition is usually a good thing. Having good intuition is almost like having a superpower.

Intuition is based on your feelings and conjecture is based on your knowledge.

Intuition is like your gut instinct, it is an explainable feeling. A conjecture is basically a formal guess, and usually has some evidence to back it up. Hope this helps!