Why is cyclooctatetraene non planar but the cyclooctatetraenide anion planar?
Fulfilling Hückel's rule means that you get to be aromatic.

The reason cyclooctotetraene is non-planar is because its planar arrangement would be antiaromatic. It twists to avoid it, because antiaromatic compounds are unstable with respect to localisation of the π-electrons (see here: What is the justification for H&#252;ckel&#39;s Rule?)

The veritable nanosecond that a molecule has the potential to become aromatic though, it will leap at the opportunity. Any angle strain from being planar is so quickly overtaken by the huge jump in stability that aromaticity brings that the molecule will twist to form an aromatic structure in a blink.