Why is cyclopentadiene more stable than its aromatic counterpart cyclopentadienyl anion?
This is a nice question because it forces you to step back and think rather than fall into the obvious trap.

You are entirely correct that aromatic compounds are more stable than non aromatic compounds which are more stable than anti aromatic compounds assuming that all other things are equal. In this case, all other things are not equal because one of the species is neutral while the other two are charged. Having a positive or negative charge on carbon is not very favourable, even if the negative charge does make the species aromatic. The $\text{p}K_\text{a}$ of cyclopentadiene in DMSO is 18[1] which means the equilibrium constant for dissociation of cyclopentadiene into the cyclopentadienyl anion and a hydrogen ion (which will actually be attached to a DMSO molecule) is $10^{-18}$! In other words, cyclopentadiene is much more stable than the cyclopentadienyl anion.