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.
PILLOW, MIST, FLEECE, and VELVET 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
Photo terms
SHUTTER / LENS / FOCUS / FLASH
SHUTTER, LENS, FOCUS, and FLASH all fit the category "Photo terms". A camera owner knows 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 keys
MAP / PIANO / HOTEL / LOCK
MAP, PIANO, HOTEL, and LOCK all fit the category "Things with keys". Each one can be associated with keys. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Desert words
DUNE / CACTUS / OASIS / MIRAGE
DUNE, CACTUS, OASIS, and MIRAGE all fit the category "Desert words". Think sand and heat. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.