We've done both.
Our first group of chickens were inherited from friends moving that couldn't take the chickens. They were older (the chickens, not the friends... well, they were too, but not relevant), but were used to being around people and being handled by kids so were still pretty friendly as pets. Soon thereafter, we added five chicks - bought at probably a few days old. The kids absolutely freaked for these - cute as hell. The chickens that resulted ended up being noticeably more attached to the family and easily handled. The downside is it takes 6 months until they produce any eggs plus all the cost and difficulty of raising them and integrating them into the flock. Another year down the road, we decided to skip the difficulty of raising chicks and bought four roughly four month old pullets. Upside - easy and cheap. Downside - it took many months for them to warm up to the kids and allow themselves to be handled.
We're now one year past buying started pullets and two years past buying day old chicks. In hindsight, the day old chicks were cute as hell, but a pain in the ass. The started pullets were less friendly for many months, nowhere near as cute, but very easy, inexpensive, and started producing eggs relatively quickly.
Down the road, however, things start to even out. The former day old chicks don't get the kind of attention they used to since they are not the stars of the show, and while still friendly, don't practically crawl in your lap like they used to. On the other hand, the started pullets that began their careers afraid of the kids now are very friendly, follow us when we walk in the yard, squat, and let us pick them up without too much fuss (differs a little between chickens).
Long story short, they both can end up similar in behavior. Chicks are more work and take longer to be productive, but more immediately bond with the family and want to crawl into your lap. Started pullets are easier to quickly get into your flock and producing eggs, but will take awhile to integrate and become socialized with the family. Both will likely end up about the same where their behavior is governed by the amount and level of physical contact they have with humans. Chickens with you from close to birth seem more uniformly friendly where started pullets (in our case anyway) show a little more range in degree of friendliness.
In fairness, ours is not a scientific experiment. If nothing else, our five chicks were all the same breed and might show more uniformly "pet" demeanor by genetics. Our four started pullets are each different breeds, so their wider range of acceptance of handling could be breed or the fact that they weren't all exactly the same age at purchase.
Hope this helps a little.
We've done both.