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.
TENSION, FREIGHT, HEIGHT, and STONE all fit the category "Hidden numbers". Each word hides a number sound or spelling. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things with tails
COMET / KITE / COIN / MOUSE
COMET, KITE, COIN, and MOUSE all fit the category "Things with tails". Each can have or refer to a tail. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things that bloom
ROSE / TULIP / FIREWORK / ALGAE
ROSE, TULIP, FIREWORK, and ALGAE all fit the category "Things that bloom". Each can bloom in its own way. 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
FLEECE / VELVET / MIST / PILLOW
FLEECE, VELVET, MIST, and PILLOW 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.