The main difference between the LED type is the color temperature of the LED. Cool white LED's operate at a color temperature above 5000K. CRI and Neutral White LED's operate typically between 3500K and 5000K.
CRI stands for color rendering index and affects how the human eye perceives a particular color source as well as how subtle variations in different shades of color are revealed. CRI is measured from 0-100 and the higher the number, the better the light source is at rendering the true color of an object. CRI and Neutral White LED's will have a minimum index rating of 80+ and produces a light that enables colors to be shown as they should be.
CRI and Neutral White LED's generally produce warmer shades (more yellow, closer to natural sunlight) as opposed to standard or cool white LED's which lean more toward the blue end of the spectrum. Due to the longer wavelength, CRI and Neutral White LED's offer some performance benefits in outdoor conditions where air temperature is not controlled and weather elements may be present.
CRI LED's are specialized and highly advanced, so it is not uncommon for a light of the same model to have a slightly less Lumen output, yet cost more than their Cool White counterparts.