Most of the old place names and literature describing New Orleans use the term bayou. Though I think the core downtown area was originally built on [slightly] higher ground above the surrounding bayou(s). I don't know what New Orleans looked like 100+ years ago, but at least today the area generally looks more bayou'ish, though I'm sure there were also plenty of swamps here and there. Certainly you don't have to go very far to reach a swamp, but neither do you need to go very far to reach a bayou.

I grew up on the Gulf Coast, mostly further east on the Florida Panhandle, but also very briefly in rural SW Louisiana surrounded by actual swamps. I suspect people today will tend to use the term swamp for its more derogatory insinuations when describing New Orleans, and bayou for other areas when they're trying to be more charitable or simply because bayou is more apt.

FWIW, to me a swamp is very stagnant and mostly overgrown with trees, whereas a bayou tends to me more open and dominated more by grasses than trees, though you can still certainly see trees (e.g. in clusters on high spots/islands). That said, the term bayou is more often heard in Louisiana, Mississippi and Eastern Texas than in Florida or Alabama, so I've not had many occasions to ponder the distinction.

For most people bayou and swamp are basically the same thing, so it seemed like a reasonable comment to me.