"In the next week" means "sometime within the next seven days." It may be easier to understand if I use months instead of weeks as an example. Right now, it is May, so if I say "next month," I mean sometime in June. However, if I say "in the next month," I mean "sometime in the next thirty days." Those thirty days include some days in May. Right now, "next year" means 2022. "In the next year" means anytime between now and May 25, 2022. "Our store will reopen sometime in the next year" could mean that I'm planning to reopen the store in August. "Our store will reopen next year" means it will definitely not reopen until at least January. The same is true with weeks. "Next week" starts on either Sunday or Monday, depending on what kind of calendar you use. "In the next week" means anytime between today and one week from now.

In addition to Gray’s correct description of how they specify different time periods, ‘in the next week’ can only be used to describe an activity occurring in the time period. ‘Next week’ is more general. My birthday is next week. (Not ‘my birthday is in the next week.’ ) I will finish the chapter (in the) next week. (Both are ok, but they have different meanings) Next week, I’m on vacation. (This describes the whole period. So you can’t use ‘in the next week’ here)

Next week - 下个星期, eg Next week, I will be on holiday (我下个星期放假). In the next week - 下个星期的某天, eg I will complete this task in the next week (我会在下个星期之内做完这件事）. In next week is not correct grammar-wise and would not be used!