In the dictionary, joy actually is a subset of happiness; it is the extreme end of the happiness spectrum. (Happiness: "a state of well-being characterized by emotions ranging from contentment to intense joy" - Joy: "intense and especially ecstatic happiness") These definitions make it clear that, technically, one can feel happiness without experiencing joy, but one cannot experience joy without being happy.
That having been said, in order to understand a word I look for the nuances of actual usage. Generally, we say that we "feel happy", while we "have joy". Likewise, we say we "are happy", but we "are joyful". Finally, we don't "happy" as a verb, but we do "joy". (The dictionary lists joy as a verb.) That's the distinction I draw. Happy is a feeling that comes and goes; joy is something that we have or possess, something that causes us to do - to "joy".
I know the limitations of this analogy, but it is much like the distinction we draw between belief and faith – a distinction many others don’t make. Belief is something our mind and/or heart feels; faith a belief that is so strong that it drives our actions - a belief so strong that it motivates us to do. Likewise, the Bible draws a clear line between the type of love that is "felt" and the type of love that is "expressed through action" - a love so strong it will cause us to act upon our feelings - to "love someone" as a verb. Paul makes that distinction explicit in his discourse on charity.
It's good to feel, but it's better to act on those feelings.