Informally speaking, these two words are used to mean the same thing; they are synonyms. However, if you are looking for a more explicit explanation, consider the following:

Anger (or feeling angry) is what people feel when they would typically say they are mad. It is a sense of being inflamed with rage.

Madness (or being mad) is a psychological state. People who are mad tend to also be referred to as psychotic or crazy, or unhinged. Think of Mad Hatter from Alice's Adventures in Wonderland, or better yet, its sequel, Through The Looking Glass.

But to be a bit more specific mad is a feeling of a state of being.

For example

I am mad. "They will be so mad with you" suggesting that they will be mad in the crazy sense but it's mostly used in the suggested way to state

(She is mad crazy)

Angry is only a feeling that you have after something has happened, its a temporary feeling or emotion.

'Mad'  is used colloquially to mean 'angry' :  'When she finds out she'll be mad at you' ;  'When I heard that I went mad'.

Mad also means crazy, mentally unstable, as in the quote about the British writer Lord Byron - "mad, bad, and dangerous to know".

Like what the others have mention, they're very similar words with not a huge amount of difference in meaning. One, however, being that 'mad' can have two uses; mad as in angry and then mad as in crazy.

I don't think people say 'mad' as much as angry when they just mean angry. I might, for example, say "She went mad when I hold her X" meaning that she got very angry. I'm more likely to speak like that to a friend, to me it's a casual way of saying someone got suddenly angry.

I feel like it's slightly more likely someone would say, "She got angry" rather than say "She got mad", but could also be personal preference, at least where I'm from.

They do mean the same thing so it's not like there'd be any confusion when using either, not a big deal or anything to really worry about.

That's the way I'd think about it.