Is there any difference between "bailar" and "danzar"?
The meaning is the same but probably what changes is how it is applied. for example:
—Vamos a ir a bailar.
—Ella va a a clase
We (Spanish speakers) usually use "danza" as the art, something with more structure and even more formal.
While "baile" is way more informal and it is the one we use as a verb. We usually don't say "Estoy danzando", we say "Estoy bailando".
At least here in Mexico.
At least in most parts of Latinoamerica, danzar is only for artistics purposes, for instance, people practicing classic dance.
Meanwhile the verb bailar is used for casual dance, I mean whatever rhythm people are dancing (salsa, merengue, tango, etc).
This is perhaps a little different than English where you mostly use to dance for everything related with this.
If you are inviting a woman to dance in a party you better say: ¿quieres bailar?
In Chile, I've heard: "baile folclórico" as well as "danza folclórica", like in this YT video title: "4 Danzas Tradicionales Mapuches", referring to artistic dance. However, as other answers have mentioned, any dance that involves the modern soul socially is almost always called: "baile". For examples, even the people who dance the artistic cueca usually say: "Vamos a bailar Cueca mañana."
I learned to say "bailar" when the chiquillos taught me to spin the trompo: "¿Sabes hacer bailar el trompo? -"¡Sí, lo sé hacer bailar hasta en el aire!"
And the ever-present fundraiser bingo is usually a "bingo bailable".