From the meaning of expression, there is not much difference between "family happiness" and "family happiness". The former is more profound than the latter, which is in line with China people's beautiful expectation of blood and affection, and also embodies the meaning of solemnity and respect.
Generally speaking, in written language, such as writing letters, couplets, short messages, poems and calligraphy, we must use family happiness, because the written language focuses on reading, and the meaning behind reading words is more important.
In casual occasions suitable for oral expression, such as speeches, wine table blessings, telephone New Year greetings and other occasions, family fun can show a relaxed atmosphere.