Is 壁橱 (as opposed to 衣柜) only a colloquial word for "closet"?
壁橱 can refer to a small closet, a wardrobe embedded in a wall, a cupboard hanging on a wall, a chest placed against a wall or something else. As the characters go (壁=wall, 橱=case), it can be a container for anything not limited to clothes. Note, as an imported concept, 'walk-in closet' is not translated into 壁橱 but something along the line of 储衣间 (clothes storage room) or 更衣室 (dressing room).

衣柜 (衣=clothes, 柜=cabinet) means a wardrobe, either built-in in the wall or a standalone piece of furniture. A general-purpose cupboard should never be called 衣柜.

In Chinese, the distinction between built-in in the wall vs. standalone furniture is not made by first-class words or affixes. Both 壁橱 and 衣柜 can refer to either a closet or something else. 

For the similarities/differences between 橱 and 柜, please refer to @Question Overflow's answer here.