From the definitions and examples I've seen, it seems that 喜悦 focuses more on the emotion of joy/happiness, and 歓喜 focuses more on the experience of the joy.
Or am I looking too far into it?
When I see 歓喜の声を上げる, I have something "explosive" in mind, like this:
On the other hand, when I see 喜悦の声を上げる in non-religious contexts, what I would have in mind is the voice of sensual pleasure, induced by things like drugs or orgasm.
According to BCCWJ, 喜悦 is rarer and much more literary than 歓喜. 歓喜 frequently appears in light novels and casual blog articles, whereas roughly half of the examples of 喜悦 are from erotic scenes in novels, and the rest are mainly from older novels and serious articles about art/religion/mythology/etc.
In religious contexts, unfortunately I'm not sure how 喜悦 is used. Hymns and Bibles are full of counterintuitive expressions, and 喜悦 may be interchangeable with 歓喜 and may refer to strong happiness in general.
This is a good question, since I have never paid due attention to before, and when I googled it some Japanese too have same question with you.
The questioner went to ( probably a shrine ) on New Years Day and there he or she bought written oracles ( probably 2 ) and one of the oracles says
The other says
The answerer is responding as follows.
Both 歓喜 and 喜悦 denotes the happines arising from deep inside one's heart, so that they do not have particular differences.
Hope mine helps even a bit.