These both appear to mean funny or amusing and I was looking for where one was more appropriate than the other. Is забавный less comical and more amusing or entertaining while смешнoй would be something you'd laugh at a great deal?
Whenever you find that in dictionary two different adjectives are translated similarly, check how dictionary deals with corresponding nouns - sometimes it can give some hints at least.
смех stand for "laughter", "laugh",
забава stands for "fun" and "amusement". Similarly,
смеяться is to laugh,
забавляться is sort of "to fool with", "to have fun with".
Roughly, "забавный" is something moderately interesting and sort of amusing.
Imagine a movie which is not necessarily funny (may be it's not a comedy at all) but yet is amusing. Imagine a fun (but not funny) and interesting fact, like in phrase:
Забавный факт - лемминги на самом деле не сигают со скалы для контроля популяции.