Symmetry and stability of a molecule (Jahn-Teller Distortion)
The symmetry and stability are not directly related per se; is benzene more stable that naphthalene for example? Benzene is more 'symmetric' than the naphthalene but both are stable molecules; $D_{6h}$ vs $D_{2h}$ point groups.

Jahn  and Teller discovered that in a non-linear molecule in an orbitally degenerate electronic state, say $e_g$ or $t_{2g}$ in an octahedral complex, that there is an interaction between the electronic orbital motion and nuclear vibrational motion, which causes the minimum energy position  to differ from the symmetric structure. This motion lifts the orbital degeneracy and lowers the energy. Thus the lower symmetry state has lower energy.

An example might be the lengthening of a pair of axial bonds in a octahedral complex, $\ce{[CoF6]}^{3-}$ for example, however, the J_T theorem does not give any indication of how big the distortion might be, i.e. it does not predict the geometrical nature of the distortion. There is also a restriction which is that if the degenerate structure has a centre of symmetry so does the distorted structure.

note: The electronic spin of the molecule (singlet, triplet etc.) is not important it is the orbital degeneracy only that matters. The Jahn-Teller effect is a breakdown of the Born-Oppenheimer approximation which is based on separating nuclear and electronic motion.