Why is the tropylium anion non-aromatic?
The rules in determining the aromaticity of a compound are as follows:

The system must have 4n+2 π electrons
The system must be planar
The system must be completely conjugated
The molecule must be cyclic.

The cycloheptatriene anion does not follow two out of these four rules. Namely,

The anion system has 4n electrons (whereas it should have had 4n+2 electrons in it π system)
The anion system has an sp3 carbon and is hence non-planar (where it should have been planar)

Due to these two issues (second issue mainly), there is no ring current in the complete ring and so no conditions need to be considered whether the compound is antiaromatic or aromatic in nature.
Cyclopentadienyl anion also has an sp3 carbon as an anion. However the anion formed is aromatic if planar and so this overcomes the ring strain and so retains aromaticity.