If you go to the USAJOBS site and search for National Weather Service, you get about 270 results. If you then limit that to jobs that are open to the public, you get just under a hundred jobs. So, I think both sides are telling A truth.
The NWS says there are around 270 open positions. That is demonstrably true.
The union is saying there are about 700 open positions. They might think there are vacancies for which requisitions have not yet been made.
Perhaps, therein lies the discrepancy. And it speaks to the rest of the odd financing and accounting discussed in the article.