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.
TREE, TOOTH, WATCH, 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
Hidden numbers
HEIGHT / STONE / FREIGHT / TENSION
HEIGHT, STONE, FREIGHT, and TENSION all fit the category "Hidden numbers". Each word hides a number sound or spelling. 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 / SIGNAL / CHOCOLATE / MUSIC
SOAP, SIGNAL, CHOCOLATE, and MUSIC 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
Words before board
SURF / DASH / KEY / SCORE
SURF, DASH, KEY, and SCORE all fit the category "Words before board". Put one word after each answer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.