The difference is the wavelength
Light is described by wavelength, measured in nanometres. Tanning beds work in the ultraviolet range, below visible light — typically UVA and some UVB. That energy is high enough to affect skin DNA, which is why tanning carries a recognised risk profile.
Red light beds work at the other end: visible red light around 630–660 nm and near-infrared around 810–850 nm. That light is absorbed by tissue but doesn't carry ultraviolet's damaging energy, and it produces no pigment response.