Religion can affect morality in a couple different ways. One way is through general religious beliefs, such as the belief in a life after death of some kind where the outcome is calibrated to conduct in this life (heaven and hell, or reincarnation), the idea that a Being who is interested in whether people act rightly or wrongly is in control of the universe, the idea that God created human being in order for those humans to behave in certain ways, etc.
One way to understand the difference this might make for morality is to consider what it is like to deny all these claims. If you say there is no God, no life after death, and perhaps even no free will, the implication is that what whether a person is a serial killer or a great human benefactor, the final outcome of your life is to go out of existence when you die. Hence serial killers and saints end up in the same condition. In addition, on some atheistic views, we are purely physical beings, and hence our actions are all determined by the laws of physics. Given the prior positions of the physical atoms that make us up and make up the rest of the universe, when we act, we could not have done otherwise from what we did. If that is so, then it seems a little difficult to say that anything that anyone does is really their fault, or is to their credit, since serial killer and public benefactor alike do exactly what nature caused them to do.
This might lead you to think that no nonreligious people have any morals, but the fact is that they are still motivated by two things: 1) natural sympathy for others and 2) the social usefulness of moral behavior.
However, there is another kind of religious moral influence, and that is the specific teachings of particular religious groups which they take to be revealed by God. But they differ on how they interpret these. People often think of strictures against homosexuality when they think of these things. (The Bible also says "Thou shalt not murder," but this isn't, for the most part, an issue between religious and nonreligious people). Some religious groups think that homosexual conduct is wrong AND that homosexual orientation is a disorder that they should do everything the can to fix through, for example conversion therapy. The second view is that homosexual conduct is wrong, but homosexual orientation is a condition that doesn't need to be fixed, it just means that God has called that person to a celibate life. (Hence a gay person can come out as gay without being condemned just for having that orientation, so long as you don't engage in any homosexual acts). The third view is that a properly committed gay relationship, along the lines of the traditional Christian heterosexual marriage can be God's will for some people of homosexual orientation.