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.
FIREWORK, TULIP, 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
Things with tails
MOUSE / KITE / COMET / COIN
MOUSE, KITE, COMET, and COIN 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 with crowns
TOOTH / KING / WATCH / TREE
TOOTH, KING, WATCH, and TREE 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
Things you fold
LAUNDRY / MAP / LETTER / ORIGAMI
LAUNDRY, MAP, LETTER, and ORIGAMI all fit the category "Things you fold". A crease is part of the process. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.