The difference will vary widely depending on the state involved, and the sources of your income, but yes, if you filed a state return as a Resident but should have filed as a Nonresident, you probably paid significantly more tax than you should have as a Nonresident and could qualify for a refund if you amend that return.
You are taxed by the state where you are a Resident (if that state has an income tax) on all of your income. As a Nonresident, you should only be taxed on income that is sourced in that state.
When you have multiple state returns, you should usually enter the earlier part-year resident or nonresident state return first, then enter the state where you resided at the end of the year, in order to get the correct results. The state where you are currently a resident may be able to give you a credit for taxes you paid to other states. There are some exceptions; for example, Arizona credits California residents for taxes expected to be paid to California on Arizona income.
Several states also have reciprocal agreements. See here for more information.
Please also see this TurboTax tips article for more information on filing returns with multiple states.
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