That’s kind of like comparing apples and oranges. Leasing is done when a rider ‘borrows’ a horse from its owner by paying a lease fee (or free leasing, no fee lease) and all associated costs for a period of time.
Sponsoring a horse involves a party paying for all or a portion of a horse’ purchase price for someone else to ride, train and show. There are levels of sponsorship, so some help pay associated fees, while others just put up the purchase fee and don’t pay for care costs or show fees.
If the horse is really expensive, some riders create syndicates made up of multiple people to help buy a horse.
Leasing a horse- ride it, take care of it at least partially, etc.
Sponsoring- you part-own or fully own the horse, but most of the time you don't have a large role in their care, exercise, all that stuff. Someone else rides them.