They are both grammatically correct and there is no difference!

No, there is no real difference. English is elastic and things can be said in several ways that mean the same thing. Hope that helps! Love Teacher Zita (BA, TEFL)

They are both correct. Just make sure not to use a double negative in English. I know double negatives are correct in many other languages, but they are not allowed in English. For example: Correct: “I don’t like either apples or bananas.” Correct: “I like neither apples or bananas.” INCORRECT: “I don’t like neither apples or bananas.” **This mistake is very common in the Southeastern US where I live. Just a word of warning in case you ever here it.

I don't have a car or a house is alot simpler, letting people know that you don't have a car or a house. I have neither a car nor a house sounds posh.

No difference.The core meaning is the same.