Both flat irons and silk presses utilize two panels of heat to flatten and lengthen out hair. Depending on the price you pay, a straightener can have special settings, protective coatings, and even steam capabilities. A flat iron can be used on its own without products — though adding in a protectant or serum is highly recommended — and a silk press uses a flat iron among other tools and products to straighten the hair. This process often goes in an order similar to this: wash, blow dry, cut, then press (via Byrdie).
While heat damage is a valid concern, using a heat protectant and spacing out the times in which you subject your hair to a straightener or get a silk press can keep this at a minimum. In fact, in many ways, silk presses can be a safer and healthier alternative from other traditional methods of straightening hair, like relaxers, which can have negative consequences, especially if used improperly (via Teen Vogue). Whether you choose to get a treatment or DIY it yourself at home, it's a matter of choosing what's best for your hair.