Crispr can do multiple things. If you want to shutoff a mutant gene, crispr cuts DNA that introduces mutations that eventually turn the gene off. Yes, you can also cut DNA and paste in a gene sequence with Crispr to fix faulty genes that you can’t just shutoff. There are also twists like Crispr base editors that can fix a single mutation without the need to cut DNA that causes a double strand break.

Contrast that to AAV. AAV doesn’t really integrate into a genome (well isn’t supposed to in theory) - they work by creating what’s known as an episome (i.e a circular piece of dna that persists in cells and gets translated into the desired protein). AAVs can only shutoff a mutant gene if they carry a payload like siRNA/microRNA or something. AAVs never really fix the mutant gene, the episome just expresses the protein that’s not working. I suppose over the long run AAVs might not really ‘cure’ a genetic disease, because the episome will likely dilute out over time with cell divisions. You can only really administer an AAV once too because of immunogenicity issues.

Also, you could use AAVs to deliver genes that encode for Crispr, so it isn’t like they’re mutually exclusive. There are pros and cons of using either of these approaches for a gene therapy. It depends on your strategy, target population, and overall risk.