"Old" would be "vecchio" but that sounds rude to Italian ears, so people usually go for "grande" and "piccolo" when comparing ages; you can use "grosso" to explicitly refer to size. Bigger isn't mistaken though, and it's not like it doesn't work in English as well (my big brother, my little sister).
I have already written to dispute this and been accepted. According to Cambridge's English Grammar in Use, BOTH are correct. You use the object "He is older than HER" when there is no second verb, but if you use the verb you must use the subject "He is older than SHE is". As for the "grande" part, in this case it refers to age, older.
Since when does grande mean older. According to DL, my answer was marked wrong because I said "... bigger than she", and they wanted "she's" - that would not be correct in English; one would either say "... than she is" or just omit the final "is", which is implied or understood.