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.
PIANO, FORTE, ALLEGRO, and ADAGIO 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 shells
EGG / ACORN / CLAM / TURTLE
EGG, ACORN, CLAM, and TURTLE all fit the category "Things with shells". Each can have a protective outer layer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Ocean words
REEF / TIDE / WAVE / KELP
REEF, TIDE, WAVE, and KELP 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 bloom
TULIP / FIREWORK / ALGAE / ROSE
TULIP, FIREWORK, ALGAE, and ROSE 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.