Aerith's death is pretty important because she kinda saves the world by dying.
But she never intended to die. That's a big thing, too. She had no interest in being a martyr, she had so much planned for her future.

Is Cloud's breakdown more impressive in the short-term? Definitely. But Aerith's death has quite the impact as well. Perhaps the impact is just felt in more subtle ways than "well the world is smurfed now."

When you get right down to it, Aerith is easily the most important character in the party besides Cloud. Even after dying, she never stops being relevant to what's happening.