As we know, the Zeeman splitting by magnetic field causes the energy difference between |↑> and |↓>, therefore states like |T+>=|↑↑> have a different energy from others. But why states like |S>=(|↑↓>-|↓↑>)/sqrt(2) and |T0>=(|↑↓>+|↓↑>)/sqrt(2) have energy differences? In this picture, T(0,2) must be in |T0> state, and it have different energy than |S(0,2)>.
We are talking here about two-electron states, which means that we need to account for the Coulomb interaction. This interaction results in exchange energy, which means that the energy of the triplet states is different from the energy of the singlet even in absence of a magnetic field.