On one level, the two numbers 616 and 666, are the numeric values of alternate spellings of Emperor Nero’s name...so apparently both were used in different versions of the early manuscripts. But 666 was the more widely used number, and, imo, is the correct one, because it decodes on other levels to other meanings related to sacred geometry and other things.
On one level for example, in the book of Revelation, it is referred to as the “number of mankind”. This is interesting because physically, as opposed to spiritually, (in other words, our animal, or “beast” aspect) we are carbon-based life forms...and 666 is the atomic number of carbon.