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.
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 crowns
WATCH / TREE / TOOTH / KING
WATCH, TREE, TOOTH, and KING all fit the category "Things with crowns". A crown can sit on each of these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Film crew roles
PRODUCER / EDITOR / GAFFER / DIRECTOR
PRODUCER, EDITOR, GAFFER, and DIRECTOR all fit the category "Film crew roles". They appear in movie credits. 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
TULIP / ALGAE / ROSE / FIREWORK
TULIP, ALGAE, ROSE, and FIREWORK 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.