Texture is what saves neutral palettes from feeling boring. A room with five different shades of cream can be rich and layered if those creams are differently textured: a smooth plaster wall, a chunky knit throw, a flat-weave wool rug, a glossy ceramic vase, a rough oak coffee table. Same color, completely different room than five smooth surfaces.
In rooms where color does most of the heavy lifting, texture can be subtle. In neutral rooms, texture has to do the work: there should be at least one rough material, one soft material, one smooth-and-hard material, and one matte-natural material visible from any seating position. That mix is what eye reads as "rich" rather than "flat".
Texture also matters in lighting. Smooth surfaces reflect crisply (and harshly under direct light); textured surfaces scatter (which is more flattering and atmospheric). A textured plaster wall under a single warm pendant looks beautiful; the same wall finished smooth under the same light looks empty.