It looks like there's no difference to these two in actual meaning? I'm guessing it's kind of equivalent to “I was” and “I have been” but seem to be more interchangeable in French?
I think there is a slight difference in meaning or emphasis.
"Je fus" seems to me like "I am now, with my attention remaining now in the present, talking about a past and finished (accomplished) event".
"J'ai été" seems to me like "I personally was then, in the past" and "reminiscing (or developing a story) I still see in my mind's eye as if it were yesterday".
I think my answer here agrees with this one.
- I say "accomplished", and it says s'est produite dans un temps défini et n'a pas de conséquence dans le présent.
- I say "reminiscing as if it were yesterday", and it says généralement récent, et peut avoir des conséquences
If you're willing to read an essay about "English language" I might characterize them as Father tongue (impersonal history, formal) and Mother tongue (personal story, oral) as told here: Ursula K. Le Guin Bryn Mawr Commencement Address