Ma'amar Mordechai 668:1 writes regarding tefillah that if one mentions sukkot instead of shemini atzeret he has not fulfilled his obligation. Although he does not explicitly say the same regarding birkat hamazon, he does write further on that:
צריך שיאמר את יום שמיני חג העצרת הזה בתפילה ובבהמ"ז ... ומשמע דעיכובא הוא
He needs to say "et yom shemini chag ha'atzeret hazeh" in tefillah and in birkat hamazon ... and the implication is that doing otherwise prevents one from fulfilling the obligation.
Regarding a difference between men and women, R. Akiva Eger in Teshuvah 1 makes this distinction in general with regard to forgetting ya'aleh v'yavo on yom tov.
He writes that women do not repeat birkat hamazon on yom tov if they forget ya'aleh v'yavo, as they are not obligated to eat like men are, since oneg yom tov is a time-bound positive mitzvah.
Note that this pesak is not necessarily agreed to by all. For example, R. E. Melamed in a footnote here writes that no-one should repeat birkat hamzaon if they made this mistake, as it there is a 'double doubt' (some say that chag hasukkot is okay instead of shemini chag ha'atzeret after the fact, and some say neither men nor women repeat birkat hamazon if they forget ya'aleh v'yavo on yom tov, as no-one has an obligation to eat bread then (apart from the first nights of pesach and sukkot)).