아니요 means "no" and it's antonym is "예/네" 아니오 is different ending form of 아니다. 그것은 내 잘못이 아니오. If some books used 아니오 as a meaning of no, it's a misspelling.

Simple answer: You can use both. 아니요 is the more common spelling. Complicated answer: You know how people say 해요 and put 요 at the end of verbs (만나요, 봐요, 마셔요 etc.)? They used to also say 하오 and put 오 at the end of verb stems (만나오, 보오, 마시오 etc,). 아니요 comes from the speech style that uses 요. 아니오 comes from the style that uses 오. No one says 하오 anymore, so 이나오 looks old fashioned and formal. The only time I've seen it is in "Yes/No" dialogue boxes on computers, which give you the options of 예 and 아니오.