Saying "lack of care" is far too strong.
The difference is not in the degree of care, but in how that care should be provided.
Remember the saying: "Give a Man a Fish, and You Feed Him for a Day. Teach a Man To Fish, and You Feed Him for a Lifetime"?
Liberals notice poverty, attribute it to lack of money, and think that the poor would be better off if they had more money.
Conservatives notice poverty, attribute it to lack of meaningful employment, and think that the poor would be better off if they earned more money.
The fish story is relatively recent (at least in that wording), but the idea is far older.
In the 12th century, Mamonides summarized Eight Levels of Charity:
The greatest level, above which there is no greater, is to support a fellow Jew by endowing him with a gift or loan, or entering into a partnership with him, or finding employment for him, in order to strengthen his hand so that he will not need to be dependent upon others
This in turn goes back to biblical principles.
When you reap the harvest of your land, you shall not wholly reap the corners of your field, nor shall you gather the gleanings of your harvest.
And you shall not glean your vineyard, nor shall you gather every grape of your vineyard; you shall leave them for the poor and the stranger: I am the LORD your God.
— Leviticus 19:9–10
God doesn't tell the farmers to deliver a percentage of their crop to the poor; he tells them to leave a portion of their crop unharvested, so that the poor can harvest it themselves.
The same principle is expressed in the Christian scriptures:
For you yourselves know how you ought to follow us, for we were not disorderly among you;
nor did we eat anyone’s bread free of charge, but worked with labor and toil night and day, that we might not be a burden to any of you,
not because we do not have authority, but to make ourselves an example of how you should follow us.
For even when we were with you, we commanded you this: If anyone will not work, neither shall he eat.
— 2 Thessalonians 3:7–10
Both liberals and conservatives can be idealists.
Liberals think that treating symptoms will alleviate the situation, and feel that the true responsibility of care is up to them.
Conservatives think that treating causes will eliminate the problem, and feel that the true responsibility for change is on the recipients themselves.
Both care, but but in different ways.
I am not talking about capital-C and capital-L Conservatives and Liberals, but about what in many countries are referred to as small-C and small-L conservatives and liberals.
Many countries have parties that are literally called "Liberal" and "Conservative" (e.g. Canada, UK), but that doesn't necessarily mean that their policies are liberal or conservative, any more than in the US, people opposed to the Democrats want to do away with democracy, or people opposed to the Republicans want to have a monarchy.
The original question didn't name any country, so my answer assumes that it is asking about conservative philosophy, not about any country's Conservative Party's policies.
It also assumes the modern American definitions of these two words:
… American conservatism is a broad system of political beliefs in the United States that is characterized by respect for American traditions, support for Judeo-Christian values, economic liberalism, anti-communism and a defense of Western culture.
Liberty within the bounds of conformity to conservatism is a core value, with a particular emphasis on strengthening the free market, limiting the size and scope of government and opposition to high taxes and government or labor union encroachment on the entrepreneur. …
— US Conservatism
… in the United States, liberalism has come to include support for reproductive rights for women, including abortion, affirmative action for minority groups historically discriminated against, multilateralism and support for international institutions, support for individual rights over corporate interests, support for universal health care for Americans (with a single-payer option), support for LGBTQ+ rights and marriage equality and opposition to tax cuts for the rich, …
— US Liberalism