North Carolina sought to prove its dedication to the CSA, after being the last to join, through arguably the greatest effort of any of the eleven states in terms of involvment. If reduced to a neutral state permitting volunteers this could mean tens of thousands of CSA troops never join up.
There's also the rather disastrous effect on Confederate transport if every train and wagon which went through North Carolina from/to Virginia must now make a vast arch through Tennessee, not to mention that anything from South Carolina must now go in the opposite direction to Georgia first.
If North Carolina is genuinely neutral this opens up an obvious target for the Union by making eastern Tennessee a much higher priority with the goal of effectively slicing the CSA apart much sooner, leaving Virginia(and the CSA government!) isolated from the other nine states.
Also the CSA violated Kentucky's neutrality for reasons far less important than the ones they would have against North Carolina yet even if the CSA were able to occupy North Carolina the resulting Confederate forces wasted on the occupation and battles, followed by guerrilla activity and all the North Carolinans refusing to serve the CSA or actively fighting the CSA could be severe.
Assuming the CSA can even find the forces to do so, with the Union armies eagerly awaiting the moment the CSA is put entirely on a (weak) defensive in one or more theaters.
Note that if only a fifth of the North Carolinans estimated to have served the CSA in some capacity form a standing army to resist any occupation then the CSA may be forced to accept her neutrality...or suffer a self-inflicted disaster by invading.