Both send an experienced person into your business. The difference is what happens after the analysis, and it changes everything about which one you should buy.
Where consulting works
Consulting is the right instrument for a bounded question with a defined answer: a market entry study, a diligence exercise, a cost review. You want an outside view, you want it fast, and you do not need the author to still be there in six months.
Where it stops working
It stops working when the problem is execution. A recommendation is only as good as the organisation implementing it, and the organisation that could not diagnose the problem often cannot implement the fix either. That gap is where most strategy work quietly dies.
The test
Ask who is accountable in six months. If the answer is a document, buy consulting. If the answer needs to be a person, you need someone in the role.