Question

What effect did Constantine's movement of the Roman capital have

Answer 1

With now two capitals competing for resources and attention, one had to be neglected over time, and since Constantinople was the hub of all commerce from the Far East and the western world, Rome lost the prestige it once had. Also Rome. Population also was affected by this move, Byzantium (which ultimately became Constantinople) was just a village before Constantine had his "vision" to build another capital there, after the administrative workers and artisans and merchants moved there, Rome started to lose its intellectual population, leaving behind poor and uneducated people. 
City taxes took a hit, since fewer and poorer people translated to less revenue there was less money to maintain the city. Also Rome was the seat of the Catholic Church which was competing for influence with the Emperor himself, so the government did it's best to isolate him. 

But the more lasting effect was the ultimate separation of the Western (Latin) and Eastern (Greek) roman empires. This division precipitated the fall of the western half bringing with it the fall of civilization in Western Europe, bringing the Dark Ages and leaving the Church as the only beacon of civilization in Western Europe, which is why western science lagged behind eastern science. 

But this also helped preserve the Roman Empire for a thousand more years, since like the old saying goes, Rome didn't fall, it's just moved. The Byzantine Empire lasted until 1456 thanks to the move made by Constantine more than a thousand years before. 

Answer 2

A lot of people had to move with it