Some Christians believe that none of the Old Testament applies, and only restrictions that can be derived entirely from the New Testament are still valid. This is fairly popular among the more liberal Protestant churches.
Some Christians believe that all laws apply, and would like to see a theocratic state that is actively enforcing all the Old Testament laws and punishments: https://en.wikipedia.org/wiki/Christian_Reconstructionism
Jesus says he fulfills the law, and despite saying not a stroke of the old law should differ, that's a loophole to drive a truck through (on the Sabbath nonetheless).
Islam doesn't have the benefit of a second prophet coming along and saying "eh, it's all good".
You really can't stereotype the beliefs of one group of Christians (or Muslims) based on another group's.
Edit: It's odd that I get upvoted for pointing out that many Christians are not bigots, and downvoted for saying the same thing about Muslims.
Yes, my mistake. You can stereotype all Muslims as terrorists, just like you can stereotype all Christians as homophobes, or all atheists as sanctimonious jerks, or all black people as watermelon-loving gangsters. It's just, y'know, stupid and bigoted.
What people don't seem to get is that many, many theists do not choose to follow all the rules in their particular holy book. How many Christian churches now endorse gay marriage, for example? You may call this hypocrisy or, indeed, heresy if you like, but you need to understand that being a Muslim does not necessarily mean that a person, their family, or their religious community holds identical beliefs to Osama bin Laden. If you judge people based on stereotypes rather than words and actions, you will judge them incorrectly.