well there is a diffrence of course :
** "moi même" means = myself
EX: je l'ai fait moi même = i did it ((myself))
** moi aussi means = me too
EX: je resterai moi aussi = i will stay ((me too))
now you also can say "Moi de même" wich has the same meaning that "Moi aussi", but we can use it as an answer only :
let's say P1 is talking with P2:
P1: Enchanté de vous connaitre. = P1: Nice to meet you.
P2: Moi de même P2: Me too.
Hope that should help you. :)
Good luck...