If you’ve ever stumbled across a papery-looking nest hanging in a tree or tucked into a shed, you might’ve wondered: was that a wasp nest or a hornet’s, and does it even matter? Actually, it does.

These variegated, collaged-looking nests — fetching up to $250 each — are a recent fixation in interior design. Indoors, a wasp nest can look like a sculpture that both “elevates living spaces” and ...