The TBG Forums uses FluxBB, but the Scratch 1.4 archive uses PunBB. Which one did the old forums use?

Technically both.

The 1.4 Scratch Forums were based on the FluxBB framework, a remix of PunBB, a free open-source forum software written in PHP.

They used FluxBB, however as ko0d3rrr said, FluxBB is a spin-off from PunBB. The short version is that PunBB is an open-source forum system that eventually was acquired by a for-profit company, but some of the developers of the original project made FluxBB as a spin-off of what it was before being acquired to keep it independent.