An assertion simply states a point. An argument gives supporting reasons why the point should be taken seriously. The reasons, then, become the topic of mutual discussion or analysis. Opinions by themselves are not proof. A mere point of view is not worthy of belief.
Belief requires reasons.
In the same way an assertion without evidence doesn't do any work. I frequently get calls on the radio show from people who think they're giving me an argument, when all they're doing is forcefully stating a point of view. They sound compelling, but a closer look reveals an emperor with lots of bluster, but no bloomers. My job is to recognize that the roof is laying flat on the ground and simply point it out. If you find yourself stymied in a discussion, you may be looking for an argument that's not there.