To paraphrase Bill Thurston, mathematics is just a way to organize human thought. The purpose of mathematical notation is to help us understand each other.

One reason that different notation is used is that the same concept may need to be compared or contrasted to different things in different situations. For instance, I recently discussed the various division symbols with someone:

The fraction notation $\frac{a}{b}$ is most useful when simplifying equations by hand.

The long division notation is most useful when calculating exact values.

The slash / is great for computers as it works with a regular keyboard.

The 'obelus' ÷ is used on calculators because the other notations are dissimilar from +,-, and x.

I feel that trying to standardize division would be counterproductive.

Now, mathematical logic is different from arithmetic, but the same truths hold. Russell and Whitehead used the notation most helpful for pure symbolic calculation, but this may not be the best notation for writing on the board or writing a computer program.

TL;DR Mathematical notation is designed to express thought as clearly as possible, and strict standardization makes this difficult.