They were primarily intended as sound amplification tools for people not suffering from any form of hearing impairment. They were, in other words, neither branded nor intended as professional hearing aids.
However, the Over-The-Counter Hearing Aid Act of 2017 has changed that to an extent. Today, some PSAPs are now classed as hearing aids. That said, there are still a few key differences to be aware of.
PSAPs don't need to be prescribed or programmed by a professional. Because they tend to be less complex than hearing aids, PSAPs don't require the assistance of an audiologist to prescribe or program. Unfortunately, this also means that it may be more difficult to find a device that's a good fit for your particular flavor of hearing impairment.
PSAPs don't work for severe hearing impairment. While an OTC hearing aid works just fine for cases of minor to moderate hearing loss, you're going to need a prescription for anything severe. The same is true of conditions like tinnitus.
PSAPs aren't fitted. A PSAP is treated as a one-size-fits-all product. They're only available in a few basic sizes, while professional hearing aids are specifically designed and fitted for your ear. 
Some PSAPs are feature-scarce. Not all PSAPs are created equal. Some lack selective amplification, while others come with a ton of different programmable sound settings. Some are feature-rich as modern hearing aids, while others are relatively basic. It all comes down to how much you're willing to spend.

PSAPs and OTC hearing aids can work wonders for someone who isn't suffering from severe hearing loss. However, they'll never be quite as effective as a custom-tailored device prescribed by a professional audiologist. Our general recommendation is to avoid them, and instead book a free consultation with a professional.