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.
PEA, LIME, MOSS, and FERN 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
Words after moon
LIGHT / STONE / WALK / BEAM
LIGHT, STONE, WALK, and BEAM all fit the category "Words after moon". Put one word before each answer. 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 / HAIL / FOG / SQUALL
DRIZZLE, HAIL, FOG, and SQUALL 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
MIST / VELVET / PILLOW / FLEECE
MIST, VELVET, PILLOW, and FLEECE 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.