Happiness, joy, and peace are three very different things. It is possible to have them all at once, but those moments are kind of rare in the grand scheme of things. The first two words in particular are sometimes used interchangeably, and so it is that I often find myself defining the difference between them. Now I find it important to introduce that third element as well.
We each get happy over different things. I am happy when I go to Disneyland, when I find a cool new chocolate bar to review, when I finish washing the dishes, when I'm watching my bearded dragon eat her morning salad, or when I see a really great bolt of lightning. None of these are bad things. But they don't really matter much in and of themselves. Some of them are fairly superficial and unimportant in a certain sense.
Joy is a little different. Think of things like seeing a newborn in your family for the first time or spending time with people you care about (not for the sake of the place or activity but for the sake of the time together) or your first kiss with someone. See what I'm getting at? Joy starts moving outside of the self. Happiness is self-centered. It can involve others, but it's mainly about you and things happening to you. Joy is more about connection. It can still be solitary, depending. But it mainly involves a connection outside of yourself. And this would be why the phrase "the joy of the Lord:" that's ultimate connection outside of the self.
And peace, peace is certainty and assurance. Peace can exist outside of and apart from happiness; it can even exist inside sadness. Peace is greater than the worries of the current moment in time. And peace is what allows you to fully enter and appreciate those moments of joy, and peace is what helps you to be thankful for the happinesses rather than expecting the things (the Disneyland or the ownership of an exotic pet) to be things that you must possess in order to be content.
Maybe peace is being, joy is interacting, and happiness is receiving. I mean, you receive peace and joy, too, but I was trying to build a nice and neat sentence there. And it loops it all back around, right? Receiving joy and receiving peace are, as I mentioned, what can bring you back around to gratefully receiving happiness. So as I started with, the three are interrelated but don't necessarily always overlap.