How is the factor theorem different from the remainder theorem in polynomials?

They are closely related but used for different goals. The remainder theorem tells you the remainder when p(x) is divided by (x − a), and that remainder is p(a). The factor theorem is a special case: it says (x − a) is a factor of p(x) if and only if p(a) = 0. So you use the remainder theorem when you simply want the remainder, and the factor theorem when you want to check whether something is a factor or to factorise a polynomial. For example, with p(x) = x² − 5x + 6, p(2) = 4 − 10 + 6 = 0, so (x − 2) is a factor. The remainder happening to be zero is precisely what makes it a factor.