The biggest thing...you know what you are getting and can see it before you even pay for it. Puppies are great, puppies are cute and you can stack the odds in your favor by getting one with a loaded pedigree but the truth is you don't really know what you have until they are older and trained. If the pedigree told it all trainers would never wash out dogs.
I bought my dog started and she is exactly what I needed...her flaws are my flaw not hers...she's a far better dog than I've been able to showcase. Because I've lacked in the training and it was my first one. I went that route because I had 2 dogs and this was going to be a 3rd...I had one chance to get a serviceable dog because no way was I going to have 4 dogs in the house.
Now, having said all of that...you need to make sure the started dog isn't a wash out and if it is what were the reasons and will they align with your needs for the dog.
All great advise...
I'll only add you get what you pay for. A good started dog is going to cost you a pretty penny.
I personally like to take a puppy and train it up because I enjoy that. If you don't just have a desire to own and train a dog from a puppy a started dog is the way to go for all the reasons Judd mentioned.