This is more cultural than historical but I'll take a stab at it since they cancelled the Mardi Gras parades tonight due to the weather. The long and short is that you're mistaken about them not having a cultural impact - their culture is similar to but different from Cajun culture and is the predominant culture in New Orleans (which varies widely from the predominant culture in Louisiana, which varies widely by region within the state).

The settlers of the area surrounding New Orleans were the Cajuns, who came as refugees and were resettled largely in western Louisiana as farmers. Cajuns are Creoles, in the sense that they're born in Louisiana, but they have a unique culture and would likely not identify as Creole if asked since nowadays that's associated more with New Orleans (although that wasn't always the case). Many still speak Cajun French or use an English dialect that borrows heavily from it. New Orleans has incorporated parts of Cajun culture but maintains a unique culture of its own that flows directly from its original settlers. If you want to see an easy example of how different they can be even though they stem from the same cultural heritage, look up pictures of a Cajun Mardi Gras and compare it to pictures of a parade in New Orleans.

New Orleans was settled in waves - French, Spanish, American, and French again. At any given point the city was a mixture of whites, free blacks, and slaves in roughly equal proportions. The slaves had a relatively high degree of freedom, especially in the city, because there were too many of them to reliably violently suppress so a blind eye was turned to some forbidden behaviors to avoid confrontation (particularly the large Sunday gatherings to sing and dance at Congo Square). This would gradually end with the American takeover (banned in the rest of the U.S., restricted to Congo Square in 1817, and fading away entirely by 1850 or so), as would much of the free black power in the city (there was even an armed, free black militia that was considered but only very prestigious but also a key component of the city's defense against colonial incursions by neighboring empires, Indian raids, and slave rebellions). The whites and free blacks often shared French culture (the Cote de Allemandes, the German Coast, was German and there are pockets of Spanish culture as well). During the Spanish and American rule, the city has kept its French identity in many ways. If you visit, especially outside the tourist areas, you will find that it sometimes doesn't feel very American at all. A lot of people try to move here and end up leaving because it's just too much of an adjustment.

So why did the city not end up like Chicago or St. Louis or Detroit, all other American cities settled by the French but then thoroughly Americanized? Continuous, active, resistance by a core population of culturally French residents. When the Spanish took possession, they sent a governor. The French colonists chased him off and continued being French. The Spanish returned with an army under the command of a certain O'Reilly, who occupied the city, executed the ringleaders, and made it officially Spain. The Creole gentry made martyrs of the executed ringleaders, set up a parallel judicial system while refusing to use the Spanish courts to settle disputes, and shunned the Spanish socially until they assimilated. The Spanish essentially became culturally French in order to be accepted and both soon agreed that local issues were best handled locally without consulting Madrid, just as they'd not consulted Paris very often (and Rome even less, despite being nominally Catholic). The colony was ceded back to France then immediately sold to the United States and nearly simultaneously a wave of culturally French refugees fleeing the massacres of the aftermath of the Hatian Revolution arrived. When the Americans came to take over "their" city, they found themselves exiled to the outskirts and barely able to access the French Quarter at risk of bloodshed, outnumbered in a city that was culturally, linguistically, and legally French. The middle of the boulevards became the neutral ground (what's commonly called the median elsewhere) and are called that to this day.

The Americans were intent on implementing democracy and the common law, replacing the French and Spanish Civil Code (slight variations, but based on the old Roman Law). The new Americans used the democracy to enshrine the Civil Code in their new constitution, where it remains to this day. As America clamped down on prostitution, New Orleans put it all in Storyville, the biggest red light district in the country. When America banned alcohol, New Orleans had more bars than before the ban and a Prohibition agent set a record for the quickest he'd ever been offered a drink - his cabbie pulled a bottle from under the seat on the ride from the airport. Marijuana is technically illegal here still but you'll find it used openly at almost any time of day and in public spaces as if it were legal. Prostitution occurs similarly. Up until 2 or 3 years ago you could smoke in any bar and this can still drink on the street or buy drinks at a drive through bar. You can also throw a parade with a marching band for a couple hundred bucks for any reason you want or no reason at all. On any given Sunday afternoon you can join a second line parade back in the neighborhood, a direct descendent of the Congo Square dance and music tradition that dates back hundreds of years (Congo Square is now in Louis Armstrong park), right down to the fact that they happen on Sunday afternoons. There's a very laissez faire, libertine attitude about the city that you don't find elsewhere in America, an almost deliberate rejection of imposed order or efficiency.

For example, (and this is where it gets cultural) the major power brokers in the state are still from the old Creole families in New Orleans, some big Cajun clans west of the city, and a few American tycoons (Landrieu, Charbonnet, Boutte, Duplantier, Thibodeaux, the German Waguespack). The legal system is still French as far as civil matters are concerned (the Civil Code) and the whole city shut down yesterday for a week to celebrate a French (or at least European) holiday that occurs next Tuesday; an entire city of 400,000 residents plus a couple hundred thousand visitors is at a standstill for a week - no one is answering phones, sending emails, anything, because they're all down at the parades eating, drinking, and catching beads. This wouldn't happen in New York, Boston, or (regrettably, because St. Louis, Detroit, and Chicago are French sister cities who have been Americanized) Chicago. Culturally, the city is barely American this time of year and only somewhat the rest (it's rare to work all that hard or often here).

That's the lasting heritage of the original settlers. It's woven into the fabric of the life of the city in ways we, the current inhabitants, almost don't understand until we try to live somewhere else. It's in the laws, the food, the music, the names of the streets (Dumaine, Duffosat, Orleans, Prieur, Pauget, Bienville, and the ones named after Napoleon's great victories, just up St. Charles from the street names after Napoleon himself - Jena, Marengo, Milan). The city is really just the culture those settlers built as they came in waves until 1810 or so, fused with bits of everything that came after.