The new product continued to be sold and retained the name Coca-Cola (until 1992, when it was officially renamed Coca-Cola II), so the old product was named Coca-Cola Classic, also called Coke Classic, later just Coke and for a short period of time it was referred to by the public as Old Coke. Many who tasted the reintroduced formula were not convinced that the first batches really were the same formula that had supposedly been retired that spring. This is partially true because Coca-Cola Classic differed from the original formula, as all bottlers who hadn't already done so were using high fructose corn syrup instead of cane sugar to sweeten the drink
What Coca Cola actually ships from Atlanta is the syrup, unsweetened. The bottlers add water and sugar to this, and AFAIK [1], it is up to their discretion what type of sugar to use.
This is one of the reasons that Coke tastes different in different places - a can of Coke in Montreal is very different from one in Toronto.
I'd have fact-checked this, but for some reason, Wikipedia is down... :)
[1] Perhaps this is no longer the case, but it certainly was in 1989, when I "interned" at a Coke bottling facility in Harare, Zimbabwe for a week