- feliĉo (lat. felix) — happy as in: lucky, being favoured by fortune, fortunate
- ĝojo (en. joy, fr. joie, lat. gaudere) — feeling joyful, rejoice, take pleasure in, being delighted
Basically being happy vs. feeling happy. The obvious connection being, that one might lead to the other.
I'm not a native English speaker but my understanding is that being glad is less intensive than being happy, e.g. I'm glad you made it. (it's good you are here, but it's no big deal) × I'm happy you made it. (it made my day that you arrived).
BTW feliĉa always trips me, because I go straight to lucky.
When "glad" is used in little conventional phrases, it doesn't mean much, you're right. But on its own, it's a strong, intense emotion -- it's what you feel when you see a favorite friend or relative, perhaps unexpectedly or when you didn't expect to see them for a long time, or when you get really good news when you expected bad news about someone or something you care about. Glad will make you feel happy all day, but you're happy because you keep remembering the thing you're glad about. You can be glad about smaller things, too but that's the gist. Happy is lower key and longer term. You're happy because your garden is growing well, or because life is just good, or because you're with someone you like, or because your cat is all warm and purring on your lap.
I subscribe to the theory that we need a reason to be sad, and, as a consequence, we can be naturally happy. In a similar way, we can be happy about something because all of the possible reasons for being sad have been avoided or have not happened. Thus, we can be happy about our work, or happy about our family, or happy about the weather being nice enough for a picnic. On the other hand, joy is closer to being in a state of bliss. An intense feeling of love could be called joyous. As a meditator, when I reach a mental state where I am experiencing my sensory surroundings without naming or thinking about what and where my various sensory experiences are coming from, nor judging any of them good or bad, I can reach a state of joyous bliss that naturally arises from the raw experience of reality. That level of meditation is more than joy--it is a sublime natural bliss--but joy is similarly intense.
"ĝui" means "to delight in (something) / to enjoy (something)" and is a transitive verb (in that it takes something as an object to enjoy or delight in).
"ĝoji" is derived from the adjective "ĝoja" which means "happy / glad." This results in "ĝoji" meaning "to be happy or glad" (but not necessarily in relation to another object).