What is the difference between 餌食{えじき} and 獲物{えもの}?
Both mean "prey", so how do we decide when to use which?
What is the difference in their nuances?
This is another one of those situations where depending on a simple English translation will cause you to lose a lot of important details.
If you look at the kanji the difference is pretty clear. 餌食 combines 餌{えさ} and 食 of 食べ物. 餌 means animal feed, and 食 is of course eat, so 餌食 becomes prey in the sense of something that an animal eats for its food, like a lion hunting its prey. It can also share the meaning of prey in English's use, as in "falling prey" to something, like being the prey of online scammers or something.
I don't know how accurate it would be to call 獲物 "prey" because I don't think that it really contains the same connotations, at least not as closely as 餌食 does. 獲物 would be better translated as "game" or something that refers to you get from hunting or fishing. The 獲 is the same as in the word 獲得{かくとく}, or acquisition. It can also be something like "spoils" in English, like in the phrase "to the victor go the spoils," referring to something like what you gain in war.
To cite my best friend, the dictionary, the two definitions are very clear:
(1)動物の餌として食われる生き物。えさ。
(1)狩りや漁で得た物。 「逃がした―は大きい」
Moral of the story: consult J-J dictionaries as much as you are able.
As ssb explained, 獲物 means "game", or it should be translated as "target" especially in context other than actual hunting of animals. 獲物 is an target, so the subject not necessarily be weaker than speaker, or maybe even tougher. Also, 獲物 has nuance in uncertain or future, a state of being targeted rather than target already captured.
On the contrary, 餌食 may be translated as "victim", indicates a state of being already hunted or weak enough and certain to be hunted down.
So in most situation of comics or novels, "こいつは俺の獲物だ" just means "He's mine (to fight)", and his hunting is not started or completed.
"こいつは俺の餌食だ" is close "He's an easy target", however, this sentence itself sounds little weird since "餌食" belongs to the past as I explained in above. "こいつは俺の餌食になるしかない(He has no choice other than become my victim)" or "こいつは俺の餌食も同然よ(I'm sure that I already have him)" is more preferred.