7. Spoken and written language (mild geek warning)
The heptapods use two different languages to speak and to write, effectively doubling their communication potential in the encounter - did this convince you?
The best Planet Earth analogy I can think of is the use in certain ancient languages (eg ancient Egyptian or Akkadian) of written markers called determinatives. These indicate some quality of the word (eg "made of metal", "is a herd animal", "is a god") but so far as we know had no corresponding indicator in the spoken language. About the only similar feature which has survived into modern English would be capital letters to indicate a sentence start or proper noun - these help the reader to parse text, but do not have any analogue in spoken conversation.
Is there any connection with the extensive use of visual icons in computer and phone screens? (or road signs, for that matter)