The word "גינה" is used more for small gardens, like your house's garden or a small garden in which you grow vegetables ("גינת ירק"). The word "גן" usually refers to a bigger garden and is also used for kindergarten ("גן ילדים").
I think גן is hardly used anymore as a general term for "garden" (as said, there is גן ילדים which may be just גן.) A garden is גינה if it's too small to be a "park". However, public gardens that have names are often named with גן. For example, גן ידידיה.