It is difficult, nay, impossible to capture that one defining spirit of NYC. When they attempted to find one by getting, arguably, the three best NYC directors (Martin Scorcese, Woody Allen and Coppola, may be they should have thrown in Spike Lee and Sydney Lumet as well into the mix to see what would come of it) together to make New York Stories, what the audience received was three distinctly different voices of the city.
You have stories of love and despair, Breakfast at Tiffany's, Working Girl or even Dog Day Afternoon (after all Al Pacino wanted to pay for sex-change operation for his lover). 
Of crime and punishment, Godfather, Goodfellas, New York, New York, Gangs of New York and a legion others. The classic stories of an outsider in NYC, fighting to survive, battling odds in the face of the city's unyielding efforts at dehumanizing him like in Taxi Driver or Midnight Cowboy
The entire oeuvre of Woody Allen where the city is the muse, a lover's embrace when things seem to go wrong or a friendly shoulder to lean on. Or finally the joie de vivre of streets in Spike Lee's Do the Right Thing and other movies which reflect that enduring spirit in the face of adversity that NYC is famous for. NYC is a kaleidoscope and aren't we glad that so many talented film-makers have created lasting stories and images twisting and turning it. 
Paris doesn't suffer from multiple identities like NYC. It has a central theme is love. You can choose any genre of film-making, the moment you set it in Paris, love seeps through the nooks and crevices, under the doors and engulfs your storyParis Je T’aime (Paris, I Love You) which had eighteen 5-minute arrondissements in Paris demonstrates this aptly especially the last story directed by Alexander Payne of a lonely middle aged American woman who comes to Paris and falls in love with no in particular. Just love, no reciprocation, no acceptance or denial. 
True love, as found in The Last Time I Saw Paris with Elizabeth Taylor and Van Johnson, or, Amelie, where a recluse waitress decides to make the lives of people around her wonderful and eventually finds love. 
Various colors of love in Kieslowski's Three Colours trilogy, all set in Paris liberating oneself from past emotions in Blue, avenging love and getting equal in the black comedy White or finding fraternal love in the most unlikely person in Red. Paris has interpreted and reinterpreted love stories over ages. Possibly the new wave with Truffaut and Goddard brought in grubby realism to the images of the city but Paris has fought back to wrest the crown of the most romantic city on screen.
