A: Let's get together for lunch.
B: OK, what about next week?
A: Hmmm, no, sorry, I'm busy all next week. What about the following week?
B: Sure, that will be fine.
'Next week' means the week immediately following the present week. 'The next week' or 'the following week means the week immediately following whatever week has just been referred to.
I hope this is helpful,