In Canadian/American English, "crazy" is equivalent to "insane", but we use "mad" to mean "angry." We only use "mad" as "insane" in specific expressions like "mad scientist" or sometimes when we are trying to sound more literary or "English".

In British English they are equivalent. In American English "mad" is more often used to mean "angry".

The term 'mad hatter' dates to the days when felt hats were stiffened by using lead or other chemicals. Apparently this caused mental illness or brain damage. Top hats were the style during the late 1700's to maybe 1850 or so. I think in Lewis Carol's 'Alice in Wonderland' there is a Mad Hatter.

There are other words that mean kind of the same thing but have a certain shade of meaning. For example, mad: furious, enraged, cross. The first two describe a much less controlled state than "cross". "Cross" can almost be equivalent to "annoyed" but is a little stronger. "Crazy" - insane, in some instances 'delusional', psychotic and psychopathic which are psychiatric terms often used incorrectly.

It's dependent the conversation. I though you got mad at me. I was very mad at you. You driver me crazy. We also use crazy for someone who lost their mind.