If you live in NJ and work in NY you have to file a NJ resident tax return and a NY nonresident tax return. You pay NY tax on your income from NY. On your NJ tax return you get a "credit for income taxes paid to other jurisdictions." The credit essentially eliminates the New Jersey tax on the income that was taxed by New York, but the credit is not the full amount of the NY tax. You still pay the full New York tax on your NY income. The New York tax rates are higher than the New Jersey rates, so you are paying more tax than you would if you earned the same income in NJ.
Unless you prepare your tax returns yourself on paper, there is an additional cost for filing a second state tax return.
SDI and SUI are paid only to the state that you work in. You only pay what is withheld from your pay. You do not have to pay additional SDI or SUI on your tax return.
New Jersey also withholds FLI. New York also withholds PFL.
SDI, SUI, FLI, and PFL are fairly minor compared to the income tax.
