I would also say as a native that "la semana que viene" is the option that most people use as a generic "next week". Then, as a synonym to that, you can also say "la próxima semana", although this sounds just slightly more formal; this is just a question of style, habit...
When we use "la semana próxima", we're generally talking about "the week after (another date we mentioned)"; for example:
Este viernes tengo fútbol y la semana próxima tengo tenis.
This Friday I have football (practice) and the week after I have tennis.
In this case, you can also use, as David said, "y la semana después/siguiente tengo tenis."
Nevertheless, it is also possible that people use "la semana próxima" to refer to a general "next week". That'd be something that would be hardly noticed by the listener. So we can see that there is a lot of flexibility here.