For federal income tax purposes, both spouses are considered to own the home, even if the home is only in the name of one spouse. That means that if you file separately ("married filing separately"), either spouse can deduct the mortgage interest and property taxes, even if only one spouse paid them. And on a joint return, of course, it matters even less, since you are basically "one taxpayer."
The name on the deed also does not matter when it comes to selling the home. Both spouses qualify to use the capital gains exclusion, as long as they both lived in the home, even if only one spouse was listed as owner on the deed. And if there are capital gains after the exclusion, and you file separately, they can be divided any way you like.
I can't tell you anything about New Jersey state income taxes or any other implications of listing the ownership in a specific way.
