So, two things I noticed from your link.
Note: Where Federal and state law have different minimum wage rates, the higher standard applies.
Which I would read to mean the Federal rates apply in those states. (just my interpretation).
And a footnote
Like the Federal wage and hour law, State law often exempts particular occupations or industries from the minimum labor standard generally applied to covered employment. Particular exemptions are not identified in this table.
So perhaps the cumulative effect is that the state laws cover some areas federal laws don’t, and these would be the only things really affected by a lower rate?
edit: also this which is a bit clearer
Federal minimum wage law supersedes state minimum wage laws where the federal minimum wage is greater than the state minimum wage. In those states where the state minimum wage is greater than the federal minimum wage, the state minimum wage prevails.