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.
MAP, LAUNDRY, ORIGAMI, and LETTER all fit the category "Things you fold". A crease is part of the process. 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
TILE / TOKEN / DIE / PAWN
TILE, TOKEN, DIE, and PAWN 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
Things that bloom
TULIP / ALGAE / FIREWORK / ROSE
TULIP, ALGAE, FIREWORK, and ROSE all fit the category "Things that bloom". Each can bloom in its own way. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Winter words
FROST / ICICLE / PARKA / SLED
FROST, ICICLE, PARKA, and SLED all fit the category "Winter words". Cold weather brings these out. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.