In most cases, hatchery vs feed store (or farm supply store) is a case of six of one, half a dozen of the other. There are success and horror stories from both sources and in the end it boils down to what you want and personal preference. Most feed stores get their chicks from hatcheries themselves, so as far as source goes the chicks may be coming from the same place anyway. Some feed stores are set up to handle chicks better than others. Some feed stores have barriers around their chick bins so that customers can't easily man handle the chicks without the help of an employee. Others have open bins that shoppers (and their children) can just reach into at will, which can result in all manner of germs being spread and chicks being put back in the wrong bin (which could mean the wrong breed, or males from straight run bins being returned to pullet only bins). Chicks from a feed store are subject to the same shipping stress as chicks ordered directly from a hatchery, plus any handling stress at the store itself (whether that's from employees picking out chicks for customers or shoppers reaching in and handling the chicks with or without permission), plus the stress of traveling from the store to your brooder. Feed stores may or may not have the breed(s) you want and they may or may not be able to order them for you if you want one that they don't usually carry. Feed stores are definitely easier (or at least cheaper) to go through for a small order as most feed stores have a minimum chick purchase of 3-6 whereas most hatcheries have a minimum chick purchase of 15-25 with much higher shipping costs for small orders (several hatcheries do offer orders as small as 3 but you are then left paying more for shipping than the chicks).
I have had good luck getting healthy chicks shipped directly from hatcheries. Our first order was a small order and we wanted specific breeds that the local farm supply store did not have (and at the time the one that had chicks in the store had a minimum chick purchase of 25 chicks, or at least that's what the sign lead us to believe, and the other store only places one order and we had already missed it) and we did not lose a single chick. Our second order was for broilers and we lost one in shipping and two shortly after, but out of 25 chicks (and the hatchery sent a few extras) and a breed that is not always known for being hardy that's really not bad. The last order we placed was for some straight run layers (leghorns, welsummers, and easter eggers) and turkeys. We did not lose any in shipping that time either. The last time, we did lose several to coccidia after we got them home, but that was my own fault because I did not check to make sure a waterer we had previously used was cleaned properly before putting it in the brooder.