For me it's the training aspect. But it's really only an issue because I train for various competition venues.
I train service dogs, and my organization gets all of our dogs from shelters/rescues. The temperment evaluations for puppies are not accurate enough for what we do, so we only adopt adult dogs -- 1-3 years old.
It is very hard to compare organizations, since the training program and standards are different everywhere, but IMO we spend just as much time training the dogs as an organization that starts with puppies. Sometimes, too, because we don't know exact ages of the dogs, we get younger dogs - 8-12 months - and we ALWAYS spend more time training these dogs, because they lack the maturity to do the job as well as an older dog. Most of a service dog's job is to lay and chill out, which younger dogs have a hard time with! So we find that older dogs are much easier to train overall.
If you compare release rates - the number of dogs that are brought into the program only to be determined to not have the temperment for a service dog - our rate is also much lower than the rest of the industry. We release about 40% of the dogs we bring in. Organizations that breed their own dogs will release about 70-80% of the litter - and they are breeding ONLY for temperment, and do extensive socialization and early training with their puppies.
So, again, it's very hard to compare, but IMO there is no benefit to getting a puppy versus an adult, or vice-versa. Personally, I'd ALWAYS prefer an adult, puppies drive me crazy!