Both gaudium and laetitia denote joy, but appear to be used differently depending on the circumstances. What is the distinction between the two (or more) Latin words for joy?
Laetitia is a state of being, from laetus happy. Someone who is laetus has gaudium, in the sense that someone who is rich has money. It more generally should be translated as happiness instead of joy.
It should be noted that the two words became synonymous in poetry and in later Latin prose.
There's also hilaritas, which is more similar to laetitia than gaudium, and means general cheerfulness. Think of it as the beaming smile on customer service's face rather than the joy at finding a sale (gaudium) or the happiness from having a series of successful encounters at a store (laetitia).
One of the differences is the way of expressing joy. This is not the only difference, but let me focus on this aspect. The aspect of expressing joy is not always present, but my understanding is that these words have such tone.
- Gaudium is inward joy that you keep to yourself. It is about feeling something inside, not about showing it.
- Exsultatio is more like jumping around in joy. It emphasizes showing the joy to everyone — which to me at least includes a mild connotation of a shallower feeling.
- Laetitia is somewhere between these two.
These words can express a difference in the way of expressing joy, but this aspect is not the only one and may well be entirely absent in some cases. As Alex B. commented, Hermann Menge (Menge 2011, Lateinische Synonymik) says basically this. Sometimes the words are used indiscriminately to mean "joy", and sometimes rhythmic considerations lead to preferring one word over its near synonyms.