These two verbs are often confused by non-native English speakers. Let’s look at the differences.
Both of the verbs involve the idea of having an intense conversation. So, what’s the difference?

According to the Oxford-University-Press-Dictionary:

argue(verb)
1 [v] ~ (with sb) (about / over sth) to speak angrily to sb because you disagree with them.

So, to argue means that there is an element of anger or disagreement involved the conversation.

example: My brothers are always arguing. We’re always arguing with each other about money.

discuss(verb)
1 ~ sth (with sb) to talk about sth with sb, especially in order to decide sth: [vn]

example: Have you discussed the problem with anyone? I’m not prepared to discuss this on the phone.

And, to discuss means that your conversation is friendly and often involves making a decision or learning more about a topic.

Does that help? Don’t forget to mentally practice these words.

Image: nuttakit / FreeDigitalPhotos.net