In hatchery birds, there is little or no difference. Hatcheries are concerned about egg production far more than they are about preserving SOP standards. As a result, their red gene birds are usually marketed under three different labels depending on their shade of red. If they are a darker shade of red (closer to mahogany), they are marketed as Rhode Island Reds. If they are a lighter shade of red (closer to chestnut), they are marketed as New Hampshires (or sometimes as New Hamshire Reds, a totally fictitious breed title). If their shade of red is somewhere down the middle or very uneven, they are marketed as Production Reds. Your bird could have been marketed by a hatchery as either a Rhode Island Red or a Production Red. If you want a comparison, below is a pic of a heritage Rhode Island Red which has a much darker red feathering, much more profuse black tail feathers, and a much deeper body than hatchery quality RIRs.
This might be another silly question, but is it kind of "looked down upon" to get hatchery chicks, or is it perfectly commonplace? I'm more experienced with dogs, so I guess a better way for me to put it is in dog terms. Are hatcheries seen as more like the puppy mills of the chicken world, or are they free of any negative stereotypes such as that? This might be an off the wall question, I'm just curious what the norm is for chickens, whether most people order theirs out of a hatchery magazine or if most have a different way of acquiring chicks, especially if they're looking for well bred chicks like the heritage one in the picture. No offense or ill intent is meant whatsoever if this happens to be a sensitive subject at all, I'm just genuinely curious and inexperienced.