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.
KELP, WAVE, TIDE, and REEF all fit the category "Ocean words". They belong near salt water. 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 fly
JET / EAGLE / DRONE / KITE
JET, EAGLE, DRONE, and KITE all fit the category "Things that fly". They spend time in the air. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Musical directions
PIANO / ADAGIO / ALLEGRO / FORTE
PIANO, ADAGIO, ALLEGRO, and FORTE all fit the category "Musical directions". A score might tell a player 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 with crowns
WATCH / TOOTH / KING / TREE
WATCH, TOOTH, KING, 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.