While one does typically hear the word illiterate to describe an individual who can neither read nor write, there are other possibilities. I've also heard the word unlettered to describe such a person, as well as unread or unschooled. If you use a thesaurus to search for words which are similar in meaning, you might also find words like ignorant, rude, or simple. I would strongly discourage you from using these words. All of these words have incredibly negative connotations and would imply that the person who cannot read nor write is stupid, and this is an inappropriate and offensive assumption.
If someone cannot read or write, it may because they have some kind of learning difference which went undiagnosed and for which they never received assistance. In all likelihood, a person who cannot read or write has had to pretend that they, in fact, can, and such a ruse might feel both exhausting and shameful.