The MM states:
A monster is defined as any creature that can be interacted with and potentially fought and killed. (MM 4).
However, despite this apparent distinction, both the MM and DMG seem to treat the two terms as synonyms of each other. Even PHB spells such as hold monster state:
Choose a creature that you can see within range... (PHB 251).
Emphasis mine. Are the terms monster and creature actually synonyms? Or is there a difference that the manuals are just ignoring?