Let's not forget that the creation of a certification program was a strong influence in propelling Scrum ahead of other ideas like eXtreme Programming. Businesses love things that make their employment selection process easier (like how resumes are often filtered based on keywords) and certifications offer them the same sort of filtering (just as college degrees do, regardless of field).

Specifically, what does Scrum offer that is not present in other approaches to implementing an Agile approach?

I don't ask this to denigrate Scrum, but to identify it's unique contributions.