1-We were going to work on the project the next week.
2-The next week was particularly hectic.
3-We were going to work on the project the following week.
4-The following week was particularly hectic.
In which case:
a-We are talking about the calendar week (the week starting on Sunday or Monday)
and in which:
b-We are talking about a period of seven days starting one day after the day we are focused on.
As written, both mean essentially the same- A. It could easily be assumed that projects would start at the beginning of a week. If we insert the word for or over in A between project and the (...work on the project for/over the next week), then:
We should also recognize that these sentences would not stand alone. Either would be part of an explanation as to why we are not working on the project in the time frame mentioned, and that context would probably clarify the matter.