Let’s talk about the real underlying philosophy for a moment.
Republicans
Republicans tend to believe that traditional institutions which have proven to be effective in maintaining a stable society should be defended. They believe the civic life revolves around the family and the community. They believe that charity should be voluntary, and that the control over a community should be strongest at the local level and weakest at larger levels, so that strangers to not have the power to compel their behaviour.
Republicans believe that the family unit is the core of American life, and that government interference weakens family bonds by reducing the need to support grandparents, reducing the need to invest in your children’s education (and for their seeing you invest in it), and reducing the need for extended families to stay together for the purpose of being a social safety net for each other. They see the rise in divorce, the rise in the number of single parent families, and the apparent weakening of ties to extended families, and they blame government policy for doing this. Especially policies since the New Deal which have greatly explanded the role of government.
They believe that ‘social engineering’ is fraught with danger, and they see much of it being carried out by people who have values inimical to theirs. They believe there are unintended consequences of meddling in the natural order of things, much as an environmentalist might oppose genetic engineering.
They are typically believers in natural rights - that our rights are intrinsic, and not bestowed upon us by government. Therefore, the government needs strong justification before taking them. These include economic rights, and most importantly the right to property.
Republicans tend to believe that hard work is rewarded, that you have to earn your living, and that living off of charity should be a temporary, last resort while you desperately try to find a way to support yourself again.
Republicans tend to trust the market, and believe that free exchange is the best way to maximize freedom and ensure that people’s needs are met to the best of our ability. They believe in the invisible hand.
Republicans tend to respect businessmen and people who invest in business.
Republicans tend to value their culture and wish to protect it. They prefer a melting pot - immigrants to their country are expected to adopt their values.
Democrats
Democrats tend to respect the worker, and see businessmen as a threat to the worker if not kept in check.
Democrats tend to believe that there is injustice in the world, and that government can be a positive force in ameliorating the effects of injustice or removing it entirely.
Democrats believe that poor people have been less fortunate than rich people, and therefore rich people have an obligation to help them.
Democrats are distrustful of the market, because they see inequality of outcome - some people benefit more than others - and they find something distasteful about having to bid and bargain for things like medical services.
Democrats tend to believe that a society works better if there is a smaller gap between rich and poor, and that it’s worth taxing the rich and giving to the poor to achieve this result.
Democrats do not trust markets to regulate themselves, and abrogate that duty to the government, which they feel is in the control of all the people, and not just the ones with money.
Democrats prefer a cultural mosaic to a melting pot. They believe that cultures are not better or worse - just different. Therefore, they support multiculturalism.
Democrats tend to be more protectionist than Republicans. In part because of their mistrust of markets, but also because they find the gap between rich and poor in international transactions to be offensively high. It bothers them that their products may be made with 50 cent an hour labor.
Democrats tend to believe that society can be adjusted and tweaked to make it fairer, and that ‘scientific’ planning of industry can result in better societal outcomes than a market which has too much influence by the wealthy classes.