I think Leanne covered it pretty well. I think of it as: Angry obviously means angry. Upset means feeling any strong emotion that causes you to feel disturbed. These could include anger, fear, sadness, etc. You original question is good, except that you can't say, "are both these words mean angry". Here's a corrected version: What’s the difference between getting mad and getting upset? Do both these words mean angry or is there a slight change in meaning?

1). It depends on the context. To get mad = always means to be angry. To get upset = - meaning 1: can mean to be sad (like you want to cry). You cannot use mad instead of this. - meaning 2: can also be used to mean angry (when you are sad/upset about something and also angry/mad about it). You could use mad instead of this. So you can usually use “to get upset” instead of “to get mad”. But you cannot always use “to get mad” to mean “to get upset”. Really Angry = mad Sad = upset Sad and really angry at the same time = upset 2). What’s the difference between “to get mad” and “to get upset”? Do they both mean to get angry or is there a slight difference in meaning?

"To get upset" is more general. I always use it to describe multiple negative emotions as a reaction to something. Not only angry, but also maybe extremely worried, anxious, or sad.