An argument is usually a verbal disagreement between two people; it may or may not be something serious. The word is always used as a noun.
*The couple got into an argument over which movie to see. *The argument got pretty serious. I could hear the Manager shouting.
A dispute, unlike an argument, is something which can last for a very long time and is frequently marked by heated exchanges. It is something serious, and takes place between two parties — they can be two organisations, countries and individuals. The word can be used as a noun and a verb.
*The dispute between the two countries has been going on for several decades. When you dispute something, you are raising serious objections. You are stating very strongly that you are objecting to it. *I dispute the claim that we have the best cricket team in the world.