I currently been reading a bit about Grothendieck topology in the book "Introduction to Etalé Cohomology" by Tamme. I've just gotten to the part where the author define the sheafification via the functor $$ (\_)^+ : \mathcal{P} \rightarrow \mathcal{P} $$ which takes an abelian presheaf $$ F : T \rightarrow \text{Ab} $$ to a seperated presheaf which in turn is taken to a sheaf by applying the functor again. My question naively is what stops us from defining sheafification of (abelian) sheaves similarly as done e.g. with compatible germs (as in Vakil's notes)? Is there just more generality in this construction?
Thank you very much for any help :)