Comparing Aegina with Hydra:
Aegina is larger with far more to do, see and experience. It has a lovely main town on the waterfront, an abandoned Byzantine Village in the hills, one of the best preserved temples in Greece, ancient olive groves, is the Pistachio Capital of Greece, seaside villages, decent beaches and is much quicker to get to than Hydra.
Hydra (Hydra Town) is one of Greece's loveliest towns with houses cascading down the hill to the waterfront. Very upscale, expensive and frequented by jet setters, partyers and people-watchers. Not known for its beaches, no archeological sites, good hiking in the interior, and if you want to escape the crowds along the waterfront you can walk uphill to traditional white-washed and pastel colored houses, narrow lanes and a bit more "Greek". It's best known for not allowing any vehicles so only donkeys can transport people and goods.
Nothing wrong with Hydra but if it was me I'd go to Aegina.