Distinguishing claims vs evidence can be difficult sometimes. They are both very similar except that claims are assertions without any factual support while evidence is an undisputable fact. Obviously, one would want evidence to be able to prove a point or at least be credible. This is an important difference because just stating claims will get you nowhere in an argument. Statements that can't be backed up are nothing more than lines of text that take up space on the page.
In the case of my own academic writing, this distinguishment can be the difference between a research paper and a fiction paper. I could make claims about my writing all day and not get anywhere. In order to make sure they are using credible facts, one must be sure that they clearly explain the evidence and show how it relates to the topic at hand. This is very important when dealing with secondary sources so that the link between the topic and the evidence is seamless.