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.  

I believe the word you are searching for is illiterate.  The word is from the Middle English and is found in Latin as illiterates.  The word illiterate can be traced back to the 15th century and means having little or no education.  According to the dictionary it means: “unable to read or write.”  Some of the synonyms for illiterate are illiterately and illiterateness.  The condition of being illiterate causes poverty and lack of self-esteem.  At one time in our country most people were illiterate, but in the 21st Century there is no excuse for anyone to be illiterate.  It is a moral responsibility from one human being to another to make sure that people learn to at least read and write.