I'd like to know what's the difference between the two because both should mean "want" but there are exercises that use one and correct the other as a mistake, and viceversa. I guess it could be that one is used for "needs" and the other for "desires" maybe?

I didn't realize there is a different until I google this just now :D So "ingin" is used for something that you want but you don't actually do anything about it, or at least not yet. "Mau" is also something you want and you are ready to do everything possible to make it happen.

Example:

English sentence: I want to be a president

Saya ingin menjadi seorang presiden - here you say that you have a dream / wish of becoming a president, but it is just in your head right now and you are not going to do anything about it, or not yet have a plan for it.

Saya mau menjadi seorang presiden - you maybe already offered to become a president, you have considered it and you want to do it, OR you are in the process of making yourself to be a president.

And since Indonesian uses slang all the time, "ingin" is rarely used in casual conversation and instead we use "kepingin" or "pingin" or "pengen". We use this word to express if we have something that we want in our mind, but we are not sure if we should have / do it, or we don't have plan to make it happen. For example you are in a diet, you have finished your dinner but you want something sweet for desert, but you are not sure if it is a good idea, but you are still considering it, you can say "aku pengen makan kue" (I want to eat cake). This word is also used by women to send hint to their love partner so this love partner will get them what they want.