Because "scrivono" is uniquely third person plural (they) and "scrivete" is uniquely second person plural (you).
what is the difference between saying "loro scrivono" and just "scrivono" since both are right?
both are equally acceptable. SInce the verb conjugation gives you already all necessary information, the personal pronoun is optional (and often left out).
Yea, from what i know from my Spanish (which is excellent for a 15 year old American teenager) the "loro" (or in Spanish, ellos/ellas) is left out 99% of the time.
That's also absolutely true for Italian as well unless your intention is to stress the importance of the person.