being fun is an instance and being happy is a feeling which u do feel within u.
being fun can mean to do something u enjoy for that instance and need to remain making u 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 u 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