The Edinburgh international festival existed before the fringe; the fringe sprang up around it. However, now the fringe is bigger in terms of number of venues, variety and length. The international festival only lasts 1 week (3rd -9th August). Whilst the fringe lasts for 3 weeks.
Looking at the Edinburgh International What’s on Website the offerings do not really appeal to me as seems a bit dated to me or perhapas I should say more classical. There is Waking for Goddart, Shakespeare Midsummer and classical music amongst others.
See website for more details: