basically means _to hang_ , _to hook_ , etc. while means _to set up_ in general.
is not very common, but is appropriate for primitive traps (e.g., rope traps) for capturing small animals, including the one in your story. is very common, and can be used with any type of trap, including complicated ones. For example, traps you see in action movies, where spears, bullets, or monsters come at a human when activated, would be , not .
Both can be used to metaphorically refer to non-physical traps (e.g., in a detective movie), in which case there is almost no difference.
Note that **** ("to entrap") and **** ("to become entrapped") are different set phrases, and you cannot say **** .