Well, I was bored, so I went and did some Research on Ebay, the found of all bicycle knowledge.
As used there, "antique" refers to bikes that are from the 1950's and earlier, with a few later bikes thrown in.
"Vintage" refers to bikes that are from the 1960's and newer with a few earlier bikes thrown in.
A lot of bikes are called "vintage" AND "antique", perhaps to get more hits on ebay searches.
In bicycle forums, I see "vintage" used quite a bit for bikes from the 1980's or maybe 1990's- basically stuff that isn't new and isn't in production, especially steel-frame road bikes or early-generation mountain bikes.
One line of logic goes like this: I'm not "old". Therefore, anything they had when I was a kid can't be an antique. Thus, a 1960's bike is clearly not an antique, and those new-fangled 10-speeds obviously aren't vintage. But of course, when a 20-year-old uses this definition, stuff from the 1980's is obviously vintage or antique.