My immediate thought upon seeing the question, and, I believe, one of the biggest examples of this phenomenon, is:

Class Field Theory

Almost anyone working in algebraic number theory uses the main results of class field theory regularly. However, even if many people have sat through a course going through the proofs of the theorems, very few people remember the proofs, and even fewer use them.

I recall hearing advice from various mathematicians that the most important thing is to learn the statements of class field theory, but not the proofs.

See, for example, this quote from the Syllabus to Brian Conrad's course on class field theory:

While it is somewhat instructive to know what goes into the proofs of the main theorems (e.g., to see what obstacles prevent the proofs from being entirely constructive), it cannot be said that the grungy details of these proofs are particularly relevant to using the theory in practice. Thus, in the first half of the course we will emphasize an understanding of the statements of the main results (in their many different forms) and will not place much emphasis on how the main theorems are proven; precise references will be given for those who wish to read the details of the proofs of the main theorems. Once we have spent some time digesting what class field theory tells us, we will study some applications of the theory, such as in the context of imaginary quadratic fields and abelian coverings of algebraic curves.