What is the best reason to use a diagram?

You've posted this under English. Is that the answer you expect? I'll deal with Math a bit later.

1. A diagram is sort of an outline. An outline is a contract between you and the material you are writing about. You are obligated to follow a set standard until you write well enough that you don't need the standard. Few people have that gift (to break the outline).

2. When you put a heading in your diagram, only that heading can be addressed. For example, if you are writing about marriage, and one of your headings is "The way Asia views Marriages based on Romance," every sentence must be the way Asia views marriage based on Romance. You cannot say anything about the way marriage works in Asia. That would be under a different heading.

3. Diagrams give you the opportunity to change your mind about importance. You can outline it one way and actually present it another, depending on how well you wrote the material.

4. Diagrams are a handy way of summarizing things. Teachers sometimes ask for an outline. Diagrams are perfect for that, but remember, each paragraph only gets 1 sentence in an outline.

Mathematics

1. Diagrams make the problem clear, especially in Geometry. In higher geometries, diagrams are never used to explain a problem.

2. Diagrams make the givens easier to show.

3. Diagrams suggest things that are not given. For example you may need a statement like AB = AB. The reflexive property is never given but it can be seen from a diagram.

4. Diagrams suggest intermediate steps.

5. A diagram is sometimes used in Algebra to show how the left side of an equality = the right side.