Hi I have different last names on my green card and in my passport,do I need any stamp in my passport verifying my identification for travelling?thank you

Depends on where you are going and who issued the paperwork. Consult your local passport office for a legal answer. I think you are going to run into a brick wall on this issue

What is the reason for the difference in names? Marriage?

Yes,marriage

You should get which ever document is in the old name, corrected

Is there any stamp or smth that confirms your new last name?

Not a problem, make sure your name on plane ticket is the same with the one on your passport. When you enter US, you would need to show green card to immigration. They have all your info on the system so it's fine.

In addition, carry a copy of marriage certificate with you, just in case. Good luck.

Thank you