Catching the Pokemon in a specific PokeBall doesn't necessarily make it better. There are 2 main exceptions:
- A Heal Ball will fully heal the Pokemon caught inside.
- A Luxury Ball starts the Pokemon caught with a higher base happiness.
Sometimes, players like using PokeBalls to bring out the appearance of, primarily Shiny Pokemon. For example, Holy Name*, used a Dive Ball on his Shiny Ralts to bring out the Blue color (I believe). Other than that, there's not really much.