I have rescued cats, instead of dogs, but the principle is mostly the same.
Pros of puppies
-you get more time with them, over the course of their lives
Cons of puppies
-training. It is very time consuming and can be a stressful process to house break a dog.
-the chewing. Puppies chew on everything, and have been known to destroy clothes, furniture, among other things.
-change in personality as they grow. As with any animal, including humans, personality changes as they grow. The puppy you get will probably not have fully developed temperament, and it’s possible they will develop traits that aren’t desirable. For example, jealousy (of children or other animals), unnecessary aggression, etc. A friend of mine had a dog who was young, who developed severe aggression with their baby, out of jealousy. They had to re-home the dog, because he bit the son.
-puppies have to be a certain age to be vetted, which can be an expensive and time consuming process. Most puppies don’t necessarily come to their new owners already vetted.
Pros of adults
-fully developed personality. Except in cases of abuse and neglect, frequently what you see is what you get.
-trained. Adults are typically already potty trained, leash trained, and know a certain amount of commands.
-less likely to completely destroy everything in your house.
-if you work 40 hours a week, an adult will be better able to handle that alone time.
-if coming from a reputable shelter, the animal will be fully vetted. That will include shots and spaying/neutering.
Cons of adults
-baggage. Some shelter animals come from abuse or neglect situations, puppy mills, or other bad situations. Sometimes, that can affect their behavior (aggression, startled easily, skittish or scared)
-if it is an older adult, you will have less time with them over the course of their life.