Yes, you are required to file both a New York and a New Jersey tax return. (I will give more detail momentarily). As far as why NJ income is lower than Federal and New York income, it depends. According to the following NJ website: http://www.state.nj.us/treasury/taxation/njit5.shtml, there are differences in how NJ treats income as compared to the IRS. Thus, the website itself recommends that you enter the income as it appears on your W-2. (If you read through the explanation on the website, you may know why, based off of the explanation given, that the number is different. If not, your payroll can help explain the difference because of NJ law).
Getting back to the tax returns, you will prepare a NJ nonresident return for the income you are earning there. You will do this first, because you are also required to file a NY resident return. New York taxes all of your income by law because you live there, but they will give you a credit for the taxes you are paying to New Jersey so you are not over-taxed. To prepare the nonresident return, please see the following FAQ: https://ttlc.intuit.com/replies/3302052