There is a difference, imo. I go to the movies on the weekend implies you only go to the movies on the weekend, and not during the week. I go to the movies on weekends implies that you go to the movies every weekend, or at least with some consistency. Similar, but not exactly the same.
I appreciate your comment. But I got confused because a couple sentences back I was using pelicula - movie. Now cine is movies but it's not, it's really "going to the (movie) theater." I used theater which was accepted, but they suggested movies. So, cine and peliculas both mean movies, but in a different way. I'm cool with that!
This is just one translation of the cinema, I'm from the United States. And going to the movies is a common way of saying (going to the movie theater or cinema.) I don't think the best translation is the movies either. So find your own better translation. Cinema, movie theatre, basically whatever you call the building that shows movies.
Every weekend would be : cada fine de semana. Or todos los fines de semana.