People often ask what the difference is between "Taiyoukai" and "Daiyoukai", wondering what meaning one confers as opposed to the other. The answer to this one is fairly straightforward:
"Daiyoukai" means "Great Youkai". "Taiyoukai" is not a word.
Japanese kanji can have many different readings depending on how they're used. Like how English speakers just "know" how to spell "phonetic" despite it not being spelled the way it sounds, for most common words, Japanese speakers just "know" which reading of a kanji is correct for each particular word. However, this becomes more difficult if a word is uncommon or just made-up.
In Japanese, the word "Daiyoukai" is written like this:
The second two kanji are unambiguously "youkai", however the first kanji can be pronounced "dai", "tai", or "oo". Someone unfamiliar with the word may accidentally read it as "Taiyoukai" instead of "Daiyoukai". But how do we know for sure that "Daiyoukai" is right?
Japanese has something called furigana. Furigana is used to tell the reader how kanji are meant to be pronounced by placing a pronunciation key beside the word, and is often found in childrens' and young-adults' comics, Inuyasha being one of them. So the furigana in the manga on the word "Daiyoukai" tells us this:
See the small characters alongside the big characters? Those are the furigana showing that this word has a reading of "だいようかい", or "Daiyoukai".
The "Taiyoukai" spelling is likely from someone reading the text without the furigana and making an (incorrect) guess as to how it was pronounced. The word "Taiyoukai" does not actually appear anywhere in the series.