In Mexico we use any of this verbs with clothes vocabulary, but what is the difference between these verbs?
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Here is some dialogue that can help show how they're often used. Here we have a parent guiding a child in getting ready to go to school.
Levántate, Juanito. Ya mero viene el autobús. ¡Vístete ya! ... Ay, ¿llevas [puestos] los calcetines de ayer? Te los quitas, por favor. Hay calcetines limpios en la cómoda, agarra, te los pones, muchas gracias. ... Hay que usar el abrigo, hace mucho frío hoy. ... ¡Ponte los guantes, hombre, mira, hay nieve!
Roughly, "usar" means "to use". "Vestir" means "to wear", and "ponerse" means to "put on".
"Usar" is not very used in Spain, at least not in infinitve. However, you can use it as an adejctive. "Ropa usada", the one that you have already worn, either by you or another person. In any case, you might want to give it a wash.
On the other hand, "vestir" means "to wear", and it is linked to the present. If you wear something, you're taking it with you. [IT can also be a custom, you wear a certain kind of clothers, frequently, but that's a present tense anyways.]
Meanwhile, "ponerse" doesn't carry that "present tense" meaning. It is a punctual action, not continuous, in the sense that you put on clothes, and that's all. You have already put them on, and they stay there, for you to wear.
I mean, when you finish "putting it on", you start wearing it. Wearing it is something continuous, while putting on is just at the beginning.