The way I (L1-english) have thought usage went is: "beestje" (as in "Huisje, Boompje, Beestje") is informal for animal, but "dier" (as in PvdD*) is formal.
“Beest” is a wild animal, or a savage being. It’s quite similar to English “beast”. “Beestje” is a diminutive that is affectionately applied to animals. “Dier” is neutral (not formal) “animal”.