I'm insulating my attic and garage. The garage will be finished (drywall).
What are the pros and cons of each method?
Sounds as if you're asking about batts versus blown in fiberglass. If that's the case, blown in fiberglass will fill all voids and you can vary its depth according to the size of the floor joists (if they're different sizes in different parts of your attic) and, if you so desire, you can easily overfill so the insulation is deeper than the joists and at even depth across the attic. Just don't overfill so much you block the air flow from your soffit vents.
The walls are another story. To blow in fiberglass into the wall voids you have to have an opening in every bay of the wall studs and after the insulation is blown in those openings will have to be patched. It's also not easy to blow in fiberglass into the walls because there are firestops in every bay and wiring in probably most of them so the distribution may not be even. If you're focused on fiberglass, batts make more sense for the walls because you can cut the batts and fit them around any obstructions before you do the sheetrock.
If you're talking spray foam rather than blown in fiberglass, it's done differently. You don't spray foam between the floor joists of the attic, you spray it into the spaces between the roof rafters. This encapsulates the attic space and keeps the attic temperature within 5 to 10 degrees of the house.
We added garage space to our house in South Carolina and I decided to use foam for the new area as well as the existing garage walls and, after soliciting bids from three companies for that work PLUS the attic of the main house, I checked references and went with the low bidder. A contractor I know used them all the time and gave them high marks. It's a big attic and with the spray foam between the rafters on 90+ degree days in the summer it gets no warmer than the low 80's whereas when we had fiberglass batts in the floor joists it was well over100 degrees. Can't remember (it was 7 years ago) but I seem to recall it was over 110 degrees on very hot days before the foam. In the winter the attic is close to the same temperature as the house because the heat rises through the sheetrock ceiling as well as all the voids for recessed light fixtures, etc.
Great place to use as work space if you have enough ceiling height.