I would like to know if there is a semantic difference between I care for you and I care about you.
In most contexts, there's no significant difference between for and about in OP's construction. Both will be understood to mean exactly the same thing.
In some contexts - for example, "She cared for her dying mother", it's more likely the intended sense would be that she looked after her mother, rather than was emotionally attached to her.
There are other possibilities. "Would you care for a mink coat?" could be asking whether you'd like one, or whether you'd look after it if you had one. But "Would you care about a mink coat?" would more likely be asking if you have any moral scruples about wearing bits of dead animals.