Obličej refers to "face" in a physical way (as a body part), whereas tvař can either refer to "face" physically or idiomatically (as in "familiar face", "save face" or "new face").

tvář can also be just "a cheek"

It's implied in the sentence that it's your face (a bit idiomatic). You can't have a spider on someone else's face. And in English you have to specify what face you mean (I think articles don't fit in this context).

There is nothing to fix. English idiom requires inserting a "your", whereas Czech does not.