What;s the difference between γράψουν and γράφουν;
Γράψω is used for short duration tenses. So it's used in the short-duration subjunctive "να γράψουν", and in future simple "θα γράψουν". It's a different tense of the same verb.
why can't it be 'αυτες'?
It can. It's one of the accepted answers.
Αυτοί δεν γράφουν βιβλία. and Δεν γράφουνε βιβλία. Please could you explain why the ε is present at the end of γράφουνε in one sentence and not the other?
Γράφουνε is another form of γράφουν. Αυτοί γράφουν/Αυτοί γράφουνε/Γράφουν/Γράφουνε are all correct and accepted.
Thank you. Do people say both? Is that why they both exist in writing? Is one form older than the other?
People do say both forms, γράφουνε is more commonly used in oral speech than in written speech, as it sometimes help with speech flow.
That's very helpful thank you :-)