Leona is a lot more dive-heavy, Nautilus is a lot less so. What Leona's kit wants her to do is be the frontline tank, get in the middle of the enemy team and disrupt them as much as possible. Nautilus, on the other hand, is more like a... "midline" tank. He peels for his team by disrupting the shit out of the enemy team.
It is, effectively, the main difference between their gap closers, Leona's will always go full range and bring her to the last champion hit, bring her as far as possible inside the enemy team, on the other hand, Nautilus' stops at the very first champion hit and brings them halfway the distance (bringing you the other half), this makes sure that Nautilus is closer to your team in the end, while Leona tends to end up pretty far away.
At least, that's how I see it. Their kits are very, very similar in design, but the main difference between their gap closers make their play patterns end up very different. Both want to disrupt the shit out of the enemy team, but Leona wants to disrupt the enemy in order to allow her allies to murder them more easily, nautilus, on the other hand, wants to disrupt the enemy team so that they don't have a chance to murder his backline (and therefore allow the backline to eventually murder the enemy team).