The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
MOSS, FERN, PEA, and LIME all fit the category "Green things". They commonly show up in green. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Weather events
DRIZZLE / SQUALL / FOG / HAIL
DRIZZLE, SQUALL, FOG, and HAIL all fit the category "Weather events". Forecasts warn you about these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Soft things
VELVET / PILLOW / FLEECE / MIST
VELVET, PILLOW, FLEECE, and MIST all fit the category "Soft things". They suggest a gentle texture. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Fast things
SPRINTER / FALCON / COMET / ARROW
SPRINTER, FALCON, COMET, and ARROW all fit the category "Fast things". Speed is the common thread. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.