I lived in MT 10 months and then moved to NC. Why did I have to pay the same amount in State taxes for both?

No one can see your information or your tax returns.   When you prepared the part-year returns for each of those two states, did you allocate the income for each state?

https://ttlc.intuit.com/questions/1901227-how-do-i-file-a-part-year-state-return

https://ttlc.intuit.com/questions/1901560-how-do-i-file-if-i-moved-to-a-different-state-last-year

https://ttlc.intuit.com/questions/2895983-how-do-i-allocate-split-income-for-a-part-year-state-return

https://ttlc.intuit.com/questions/2824532-i-need-to-amend-my-state-return

It sounds like you failed to allocate your income.  You are a part year resident of both states and, as such, you file a part year resident return in both states.  

In the each state interview, you will be given a chance to allocate each piece of income as that state income or the other state income. I'm not specifically familiar with the  MT and NC software, but it works that way in all states. The actual entry wording may vary.

At the W-2 screen, in the federal income section, be sure the box 16 amount, for each state matches what you earned in that state.  If your employer did it wrong, you are allowed to change it, in TurboTax, to get the result you need.