European towns and villages often have many centuries, possibly millennia, of history. That at the very least adds depth.
Few American cities were founded > 200 years ago. None were founded more than 1565 (United States), and the oldest dates from 1502, Santo Domingo, Dominican Republic.
That said, what a New Yorker visiting small-town Europe is probably referencing is the contrast between NYC and Small European Town. The specific town will likely be largely interchangeable with numerous others of similar description.
By contrast, a small town in the US or Canada largely resembles some of the less-distinguished suburbs of NYC itself: same language, same government, same mass media, same history, same chain stores and franchises. Far fewer people, however, and a much smaller economic and cultural ambit.
Even for small towns in which there is some distinctive culture ... it tends to be a single motif which dominates the entire town, rather than the melting pot of a major metropolis. Marfa, TX; Silver City, NM; Santa Fe, NM; Asheland, OR; and Napa, CA all have distinctive cultures, but it tends to be all-of-a-type.
Exceptions tend to be smaller college towns, for obvious reasons, but even here, the odds for highly-diverse communities tend to be lower than in larger cosmopolitan cities. And you can climb far up the major metro region and still find fairly stultified culture --- Chicago, IL, wears its "second city" moniker as a badge, but you'll still find there's some real edginess there about it if you press a bit or scratch beneath the surface. (The fact that it's third by size, and that Los Angeles also evinces a somewhat similar attitude toward New York doesn't help.)