Arguments and discussion both are part of any conversations, in which people talking with one another to tell what they think, feel or know to be true of any issue.
Though the conversations between people start as discussion, many times, the conversations turn into arguments, because of people's pre-occupancy with fixed opinions about the issues being conversed.
The Key difference between an argument and discussion is:
In an argument, people force to convince the others that their viewpoint is correct.
Argument is like fighting a losing battle, i.e., winning an argument at cost of losing a job, customer, friend or marriage, which is of no use.
A discussion is a conversation in which people try to understand each other's views/opinions on the issue being conversed, i.e., discussion is an exchange of views, knowledge, expression of logic, which helps to find solution of the issue.
Simply — ARGUMENT is to find out WHO is right. DISCUSSION is to find out WHAT is right.