I am interested in Nutrition. Syracuse has the perfect major for my interest, Tulane does not. The weather is a turnoff for Syracuse. Both socially I like. I have visited Tulane but not Syracuse.
See if you can fit in a visit at Syracuse. I’m used to that type of weather so that wouldn’t be a turn off for me, since it has the better academics concerning your major, I’d pick Syracuse, but only if you think you’d be happy there. You can deal with the cold.
i’d definitely pick Syracuse 
New Orleans is sketch 
I’d to go Tulane. The cold northern winters in Syracuse would not agree with me; however, I wouldn’t be present through the sticky hot summers in Louisiana so the mild winters would be more attractive. Tulane is a great size and is a more selective place, so classmates and recruiting will be different. New Orleans is such a thriving and wonderful city. College students change their major all the time, and the exact major you pick in undergrad isn’t so important (unless you want to be an engineer or a nurse). I’d just major in public health instead - Tulane has an excellent school of public health with some coursework in nutrition and professors doing that research.
As a New Jerseyan who goes attends school in the South due to disdain of cold weather, go to Tulane.