They share no other melodies...unless I'm too far mistaken. They do swell up, but Aeris' theme does it within half as many measures. Then it goes to an F chord; Aria di Mezzo does not contain this chord. Then it goes to D minor, a chord also not present in Aria di Mezzo. At this point, Aeris' theme has a clarinet (or something, an instrument that isn't present in Aria) take over the melody, and quiets way down as it modulates to F#minor (and skips a measure of beats while doing it), while Aria di Mezzo has an extra measure (with a B minor arpeggio) before modulating to B minor and bringing the volume way up again, much quicker.

I could go on, but I'd cap this off by noting that Aeris' theme repeats if left to play on its own; due to the nature of the Aria, it has an ending (rare for the FF series).

They really only share the three notes at the intro, and similar instrumentation. Also, they are both in the key of D, so I guess it's really a matter of opinion.