Natural language is a great querying tool for quickly combining existing knowledge in order to generate moderately intricate thoughts. It feels easy, but don't forget that it took many lifetimes of experience to accumulate the database you are drawing upon. It's similarly easy to throw together a Smalltalk prototype as it is to have a casual conversation. Not so easy is to construct a language--natural or artificial--from scratch (let alone try to teach it to somebody).

On the other hand, in advanced mathematics, diagrams are used all over the place (look at category theory). When they're not used, the reader is essentially forced to imagine spatially what is being said, often down a blind alley (to use the phrase almost literally).

Diagrams are essential to communicating non-obvious abstract ideas in an efficient manner. In addition, given a domain-specific technical language, there are many possible permutations of words in a sentence, whereas the space of sensibly drawn diagrams is much smaller. Often, after fully comprehending an idea that was previously expressed in words, the final step in congealing the idea in your mind (and convincing yourself of its soundness) is to draw a definitive diagram which covers all the cases, and can be understood in a glance.

I should also mention that many of the worst typographical errors in mathematics completely change the meaning and jumble the thought, something which is far more difficult to do when using a diagram.

Also, as I remarked in my other comment, diagrams complement text, with either one enhancing the other. Neither should completely replace the other.