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.
SEED, BYTE, ATOM, and PIXEL all fit the category "Tiny units". Smallness connects them. 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 / PAWN / DIE
TILE, TOKEN, PAWN, and DIE 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
Musical directions
PIANO / FORTE / ADAGIO / ALLEGRO
PIANO, FORTE, ADAGIO, and ALLEGRO all fit the category "Musical directions". A score might tell a player these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Garden tools
HOSE / SPADE / TROWEL / RAKE
HOSE, SPADE, TROWEL, and RAKE all fit the category "Garden tools". These live in a shed. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.