The regular Coca-Cola recipie sold in the U.S. uses high-fructose corn syrup (HFCS) as its sweetener. And being soda, there's quite a lot of it. According to Wikipedia on Coke's HFCS use

Since 1985 in the U.S., Coke has been made with high fructose corn syrup instead of sucrose to reduce costs. [...] Though the Coca-Cola company claims that there is no difference in taste, many people claim to prefer Coke made with sugar. [...] Kosher for Passover Coke is also made with sugar, rather than corn syrup, due to the special dietary restrictions for observant Jews (Orthodox Jews are prohibited from consuming corn during this period) during the holiday.

Here's a more detailed explanation of Passover Coke.

HFCS, like most corn products, is very cheap in the U.S. because of corn-farming subsidies and very high supply. Most packaged foods and drinks in the U.S. use HFCS instead of sucrose (regular "sugar") only because it's significantly cheaper, despite some preliminary research that may indicate that it's unhealthy and leads to obesity and diabetes. (I think this is ignoring the obvious: overconsumption of any sugar leads to obesity and diabetes.)

Many die-hard Coke fans are convinced that its reformulation with HFCS damaged the taste, claiming that the sucrose version was less "sticky" or had a better aftertaste. To test this assertion, I purchased Passover Coke and did a blind test (good idea, Tiff, and thanks for conducting it) to determine which tasted better.

My verdict:

I could tell there was a difference when I was looking for one. It's pretty obvious once you pin down exactly what the difference is. (But that's not easy.) But it's not such an extreme difference that I'd notice which one I was drinking if I was casually drinking one of them while doing something else.

I preferred the Passover Coke, even before the identities were revealed.

The Passover Coke was less sweet when it first hit my tongue, and the sweetness faded more quickly.

Regular (HFCS) Coke had a lingering too-sweet aftertaste, as if I hadn't brushed my teeth in a while. Passover Coke distinctly lacked this.

That's about it. If I was a die-hard Coke fanatic (I'm not, and I rarely drink soda), it would certainly be worth trying to find some Passover Coke for curiosity's sake. But it's not worth driving to a different state or mail-ordering it.