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.
VELVET, PILLOW, MIST, and FLEECE all fit the category "Soft things". They suggest a gentle texture. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
River words
BANK / DELTA / CURRENT / RAPIDS
BANK, DELTA, CURRENT, and RAPIDS 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
Board game pieces
TOKEN / DIE / PAWN / TILE
TOKEN, DIE, PAWN, and TILE all fit the category "Board game pieces". You might move or roll these on a tabletop. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Stage terms
ENCORE / CURTAIN / SCENE / PROP
ENCORE, CURTAIN, SCENE, and PROP all fit the category "Stage terms". A theater audience might notice them. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.