12 inch arms 9 inch arms - why?

One difference is that you will need a mahusive plinth - as fitted to my Goldring '88' - to accommodate its 3012 arm. I can imagine a few slight advantages and disadvantages but never heard an informed explanation.

Yes, not all arms are 12 or 9 but they were common lengths.
 
I believe the 12" tonearm was originally designed to accommodate 16" radio broadcast transmission discs.

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BBC 16" transcription disc

The 9" tonearm was adopted for domestic use with 12" discs.