The difference between an and année is quite subtle. Indeed, The meaning doesn't change (in both cases it means year), but the use of both words differ. Here is a simple way to make the difference :
The word an is used with cardinal numbers (1, 2, 3...) :
Il y a 1 an > One year agoJ'ai 6 ans > I am 6 years old.Elle a 23 ans > She is 23.Il a 34 ans > He is 34.Dans 100 ans > In one hundred years.Most of the time, the word année is used with adjectives and ordinal numbers (the first, the second…) :
Toute l’année > The whole year.L’année prochaine > Next yearLa première année > The first year.La deuxième année > The second year.This previous rule works in 90% of the case. But, sometimes, an and année can commute..
L’an prochain / L’année prochaine > both mean next year.L’an dernier / L’année dernière > both mean last year.Here, they have the same meaning. You can use both, although l’année prochaine and l’année dernière are more commonly used.
Sometimes, you can find a cardinal number with année :
J’ai passé quatre ans au Mexique.J’ai passé quatre années au Mexique.No meaning difference, but quatre années insists more on the duration.
Le nouvel an is January the first. Le nouvel an is celebrated on the evening of December 31st. In other words le nouvel an is kind of the New Year's Eve.
La nouvelle année is the new year that begins on January 1st. La nouvelle année is 2026, for exemple.