Then why call it the Eastern Roman Empire, or the Byzantine Empire if it's the same thing?
Sure, splitting the Empire in 286 was simply a practical matter, but by the time Constantine I moved the capitol and changed the state religion, just because they still called themselves "Romans" day to day until the Ottomans showed up in 1453, doesn't automatically mean they're the same empire. There were 2 emporors and 2 courts, effectively 2 separate governments. In very technical terms, the split was a legal loophole to maintain a "single" empire on paper for their own egos and to avoid conflict. In practice and in hindsight, they're 2 different things.
It's a bit like saying that any Commonwealth county is the same as the UK. It's not truly. Let's imagine that just before WWI, King George V decided to pick up and move to Australia and start calling it "The Southern United Kingdom of Great Britain and Ireland," and leave another family member in charge as "co-regent" in London. Then watch as WWI tears Europe apart and the German Empire 50 years later invades London and takes it and colonial countries scatter away from George V's grip. Does the British Empire still exist? I would say no. What say you? Other than this would make some great alternate historical fiction.