I'd say they are pretty much the same. There may be a slight difference in the tone. For example, you might begin a story with "Once . . . . etc." But you'd be less likely to to begin a story with with "On one occasion . . . " That's just a feeling I have, not a rule.

"Once" does not necessarily refer to any occasion. "Once" is closely tied to the number "one". For example, you can say "the letter 'e' appears once in the word 'eat'" but it would sound funny (but not be wrong) to say "the letter 'e' appears on one occasion in the word 'eat' ". As another example, you can say "you can only go swimming once this week". It would sound odd (but not be wrong) to say "you can only go swimming on one occasion this week". It would sound odd because the word "occasion" has no clear reason to be in the sentence.

When talking about the past, when 'once' means 'on one occasion', they are interchangeable. 'Once' has other meanings and 'on one occasion' cannot be used in those situations. 'Once' can mean 'when' or 'as soon as' in the future. 'Once' can also be used to mean 'at a time in the past' more widely - He was a teacher once = He used to be a teacher.