Mad-Angry | Commonly Confused Words

Mad

/mæd/

Feeling or showing strong emotions such as anger or excitement, often preventing reasonable thought.

He has to be the maddest man I've ever met.He is loose outside there now - and he has mad!'Why would I be mad?"

/ˈæŋɡri/

Feeling strong displeasure or hostility towards someone or something

Tom was very angry yesterday.Your words only make me angrier.'He is not famous and he has angry about that."

Mad can mean crazy, which goes beyond just being upset.
The British often use mad to mean angry, while in American English, it typically means insane.
Angry is more direct and simply refers to being upset.