being fun is an instance and being happy is a feeling which you do feel within you.
being fun can mean to do something you enjoy for that instance and need to remain making you happy.
whereas being happy can intend u to make happy for a longer period of time. enjoyment is different from happiness, though it is true that doing something which u enjoy can make you happy. but it depends on the duration of the activity. if it is a small duration activity then it may not leave u happy for a long time.
hope u understood