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.
TULIP, FIREWORK, ROSE, 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
Green things
LIME / PEA / FERN / MOSS
LIME, PEA, FERN, and MOSS 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
Things with tails
COIN / MOUSE / KITE / COMET
COIN, MOUSE, KITE, and COMET 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
Hidden numbers
HEIGHT / STONE / TENSION / FREIGHT
HEIGHT, STONE, TENSION, and FREIGHT 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.