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.
SUSPECT, MOTIVE, EVIDENCE, and ALIBI all fit the category "Mystery story clues". A detective asks about 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 bars
SOAP / MUSIC / CHOCOLATE / SIGNAL
SOAP, MUSIC, CHOCOLATE, and SIGNAL all fit the category "Things with bars". Bars can describe each answer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Spicy things
GINGER / PEPPER / CHILI / WASABI
GINGER, PEPPER, CHILI, and WASABI all fit the category "Spicy things". They can wake up your taste buds. 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
KITE / COMET / MOUSE / COIN
KITE, COMET, MOUSE, 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.