There's actually a big difference and a lot of nuances.
A farrier has gone to school usually but not always. They have learned some black smithing, making shoes and manipulating existing shoes. They learn corrective shoeing. They also learn to trim.
A barefoot trimmer is someone who may or may not have gone to school. They don't typically know shoes - only trimming. They're no necessarily bad.
Realize that in most states, a person can just call themselves a farrier or barefoot trimmer and open a business as one and its not illegal to do so.
Generally speaking, farriers are more educated in terms of hooves, shoes and trimming.
Now, a bad trimmer or a bad farrier can both damage the horse. Some people like to give a blanket response as to whether shoes are good or not. My blanket response is "it depends". If your horse can function in the way you want, comfortably without shoes then go barefoot if you're comfortable doing so. Not everyone is. Not all horses need shoes in all circumstances. Some horses need shoes always, some do fine without any ever. You have to do what's best for the horse. If your horse is barefoot and is comfortable then consider keeping her that way.
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