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.
KITE, COMET, COIN, and MOUSE 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.
Category
Green things
LIME / MOSS / FERN / PEA
LIME, MOSS, FERN, and PEA all fit the category "Green things". They commonly show up in green. 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
ACORN / EGG / TURTLE / CLAM
ACORN, EGG, TURTLE, and CLAM 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
Map features
BORDER / LEGEND / COMPASS / SCALE
BORDER, LEGEND, COMPASS, and SCALE all fit the category "Map features". These help you read a map. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.