I'm assuming that you mean the French influence is strong in New Orleans as compared to other territories of French Louisiana added to the United States in the Louisiana purchase? Obviously, the French influence in New Orleans pales in comparison to that in Quebec. So taking the question as why is the French influence more obvious in New Orleans than in, say, St. Louis, I think the underlying answer is population. In 1810, New Orleans was a city of 17,000 inhabitants -- nearly all of them French speaking. As discussed below, Spanish Louisiana was really a continuation of French Louisiana and another 8,000 or so French speaking immigrants arrived in New Orleans in 1809. So, 1810 New Orleans was a large, well-established, Francophone city. The other cities of the Louisiana territory were not and so did not have the same cultural persistence.

As you note, New Orleans went through two colonial powers: France (to 1762) and essentially Spain -- though with the briefest return to France to the Louisiana purchase in 1803. You should, however, really think of this as a century of French influence -- not a half century each of divided French and Spanish influence.

During the Spanish period after 1762, Spanish cultural influence on the Louisiana territory broadly was limited. The first Spanish governor arrived in New Orleans in 1766 and was driven from the city by a Francophile revolt in 1768. The Spanish suppressed the revolt (oddly enough the governor sent to do this, Alejandro O'Reilly was Irish by birth) and then installed Luis de Unzaga as governor. To prevent further difficulties, Unzaga decided to embrace the francophone Louisiana elite. Unzaga married a francophone Louisian Creole woman (Marie Elizabeth St. Maxent), allowed significant cultural and political autonomy, and allowed the francophone Creole elite to continue to dominate affairs. Spanish culture simply did not penetrate Louisiana.

New Orleans also received several waves of French-speaking immigration. After the Haitian revolution, a large number of French speaking refugees (including much of the colonial elite) fled to New Orleans beginning in the 1790s. A number of French monarchists also fled France for New Orleans during the French revolution. A number of other Francophones fled Haiti for Cuba. In 1809 many of these were expelled and fled to New Orleans. In 1809 alone, over 9,000 of these (almost exclusively French speaking) refugees arrived, roughly doubling the population of the city.

Thus, we get to the early 19th century with a large, well-developed and French influenced city, so the next question is why this didn't unravel over the next 200 years. Of course, it did to a considerable extent. Again, New Orleans is not Montreal -- it's a clearly American, English-speaking city. There was simply a large enough cultural force to have some persistence over time -- thus, the French street names, French cuisine, fondness for the -eaux suffix and so on.

In 1812, Louisiana (as a whole, I don't know of New Orleans statistics) was 75% French speaking. Thereafter, there was large scale English speaking immigration, and Louisiana (again as a whole) was 70% English speaking in 1860. The French culture was stronger in rural South Louisiana than in New Orleans. In 1852, the New Orleans Knickberbocker declared: "The New-Orleans of 1852 is not the New-Orleans of twenty years ago. The innovations of the Anglo-Saxon race have been steadily undermining the manners and customs of the aborigines of the country ... The Creole influenced breathed its last breath ... New-Orleans is now an Anglo-Saxon city." The aftermath of the Civil War dealt the death blow to Louisiana French speakers -- the Reconstruction-era constitution mandated English language education. The French speaking newspapers in Louisiana largely closed in the second half of the 19th century.