It seems to me that conservatives and liberals think differently - in politics as well as in religion. It isn't just a matter of coming to different conclusions, but we have different processes by which we reach conclusions.
Conservatives often differentiate them from us by staying that we are Moral Relativists and that they are Moral Absolutists. Sometimes they say we have no Moral Compass. However, many of our stances do not come from a lack of morality, but from seeing more gray areas, more complexity, and being more comfortable with ambiguity. Even our idealism works differently, since we're both idealistic. Often a Conservative will talk about how they wish things to be, and make policy as if the ideal were already reality. Liberals think of how things are, and the inherent problems, and then en-vision a better future and how to get there.
One issue which consistently makes the difference between whether one considers themselves Liberal or Conservative, or whether they'll vote Republican or Democrat, is the issue of Choice when it comes to abortion. Those called "Pro-Life" simply state an absolute "no". They quote various verses from the Bible that affirm Life from the womb, and God's authority over us. (Isaiah 44, 49, Job 10, 31, Psalm 139, Amos 1:13, Romans 9:20, etc.) They are adamant about protecting the life of the fetus although they are less adamant about protecting other life since they tend to be against programs that would help the child after it's born, and usually are for capital punishment and for war.
Liberals, however, look at the issue differently. A Religious Liberal might begin with Scripture, and notice that there is nothing in the Bible about abortion unless one reads Number 5:12, which seems, on the surface, to make a pro-abortion statement. They'd put the verse into the context of its time, and move on.
They might then begin to think through the problem. Ideally, yes we'd want all children to be wanted, enough money to care for children, a family that loves each other. But this isn't always workable - no matter how much we want it.
A Liberal might then read statistics and notice that countries which forbid abortion seem to have a higher percentage of abortions, (read the breakdown of statistics in Freakanomics, for instance), and more women die when abortions aren't legal.
Conservatives might tell women that they should have a child and give the child up for adoption. But many women would reply, "I'll lose my job if I do that due to the possibility of frequent sickness and frequent absences from work. I don't have money nor health insurance for the doctor visits during pregnancy."
Some Conservatives might tell her, "You should have used birth control and been responsible so you should take responsibility for your actions." She might reply, "I did use birth control but it isn't 100% effective. And why should a baby, and/or a whole family be punished because of this pregnancy? Babies are not supposed to be punishment."
The discussion with the Conservative might end there, but the Liberal might continue to think about this. Some Liberals might ask an ethical question, "Is it better to do harm, or to do good?" If a policy does more harm than good, perhaps we shouldn't follow that policy.
Some might decide to re-frame the argument, as Hillary Clinton has done. She recognized that the problem isn't abortion, the problem is unwanted pregnancy. Therefore, how do we reduce unwanted pregnancies?
Some Conservatives might reply, "If someone is moral, they won't have unwanted pregnancies?" Liberals might reply, "Not true. Many pregnancies come from failed birth control, whether with teen-agers or with families. Unwanted pregnancies come from rape as well. Morality is not always an issue here."
Conservatives preach a sex-education program of Abstinence, which is not effective, by itself. (Read Henry Waxman's study of this.) Instead, recognizing that many social problems are complex and take a variety of solutions, Liberals want a series of solutions which include an accurate sex education program that includes speaking about abstinence as well as access to condoms and other forms of birth control for those who decide to have sex - whether because they're married, or because they're irresponsible, or because they feel passionately about someone in a long-term, or not long-term relationship.
Recently, Democrats-for-Life have had a place in the Democratic Party, working with Pro-Choice Democrats to reduce abortions (see democratsforlife.com.) and trying to be effective, not just moralistic.
Liberals try to think through the complexity, recognizing that this thorniest of moral problems may need more creative thinking to find workable solutions.