"Clean the room neatly" is correct because "neatly" is modifying the verb "clean." You can also say, "Neatly clean the room." Neat is an adjective, and adjectives are used to describe nouns and pronouns. If you say "Neat clean the room," or "Clean the room neat," you are describing the verb, and you want the room to be neat, not the verb clean.

Clean the room neatly <-- correct because "neat" is adj