The total-spin singlet state for two electrons has one spin up and one down, but one of the triplet states does. too. What is the difference?

By absorbing radiation, an electron in a molecule (singlet ground state) is excited to a higher energy level (excited state). This new excited state can be singlet or triplet.

The key difference between the singlet and triplet state is that the singlet state shows only one spectral line whereas the triplet state shows a threefold splitting of spectral lines.

The total-spin triplet state for the two electrons is symmetric under the exchange of particle labels for the spins, while the singlet state is an anti-symmetric combination of spins that changes signs under the exchange of particles. Specifically, with the rust label indicating particle 1 and the second particle 2 in each pair of spins, the three triplet states, and the singlet state respectively have the spin part