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.
DELTA, RAPIDS, CURRENT, and BANK all fit the category "River words". Fresh water flows through this set. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Night sky
METEOR / VENUS / ORION / NEBULA
METEOR, VENUS, ORION, and NEBULA all fit the category "Night sky". Look up after sunset. 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
TREE / KING / TOOTH / WATCH
TREE, KING, TOOTH, and WATCH 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 with tails
COMET / MOUSE / KITE / COIN
COMET, MOUSE, KITE, 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.