Applying to Grad School for Byzantine History from St. AndrewsWatch
Need more help on going postgrad?
Oxford has more prestige than Edinburgh, esp. in America. It is not necessarily 'better' than Edinburgh, but Oxford has more impact for employers and US unis.
The Oxford course is more specialist than the Edinburgh one, which includes Islamic history. At Oxford, you do either Late Antiquity or Byzantium. At Edinburgh, you can mix and match modules from Late Antiquity, Byzantium and Islamic history. (Although that might be an advantage due to diversity of choice, it might have the side-effect of you being forced into certain modules for staffing reasons.)
Oxford also has more specialist staff than Edinburgh, although not by much.
Oxford has more focus on ancient/med languages, which you definitely need for PhD.
This might come across as very pro-Oxford, but these are the factors that I thought about when I was making the same decision.