If by 'conservative' you mean 'moderate Republican', and by 'liberal' you mean 'moderate Democrat'.
If 'conservatives' had an absolute majority, then probably:
1) Same-sex marriage would be rolled back, and replaced with civil unions that provide the same rights as marriage, which was essentially Romney's stance.
2) Flat taxes would become the norm, and taxes at the top income brackets would decrease further.
3) There would be a private insurance system, if you can't afford cover then you have to rely on healthcare from charities and state programs.
4) Social welfare will be cut back, though public education funding will probably remain the same.
5) Deporting illegal immigrants will be attempted, possibly there would be ID checks and legislation making it easier to deport illegal immigrants.
6) Religious education would be more readily allowed in public schools, and bible study would become a part of most classroom curriculums.
7) Abortions would be restricted to cases of rape, and those that could afford abortions (but couldn't get one due to strict US abortion laws) would get it done discretely in Canada, Europe, and even Mexico.
If 'liberals' had an absolute majority, then probably:
1) Discriminating against LGBT people would become illegal and would be treated like discrimination on race, also same-sex couples would be able to adopt.
2) 'Progressive taxes' would be implemented, where the higher income brackets pay the most percentage of tax, and the less you earn determines your tax code.
3) The ACA would eventually be replaced with a universal healthcare system akin to those in Europe, or at the very least a system equivalent to that in Canada.
4) Social welfare programs will be expanded, and public education spending would increase.
5) Illegal immigrants will be given continual amnesty, and social welfare and education programs would be pressured by the strain.
6) Religious education would be equivalent to what is allowed in blue states, though schools would be encouraged/pressured to include a curriculum with multiple religions where religion is taught.
7) Abortion would become more readily accessible, and would be made affordable for those on lower incomes.