What is the difference between the black body and the reconstituted daylight spectrum?

The black body spectrum is a spectral plot of a "theoretical object" that emits radiation (black body radiator) at a particular temperature. The sun is often mentioned as an object that comes close to a true black body radiator, and this makes it relatively straightforward to calculate the solar spectrum based on Planck's law.

When we discuss daylight, however, there is a distinction between the solar spectrum and daylight spectrum due to the way in which the light interacts with earth's atmosphere. Daylight spectra are essentially solar black body radiation with certain wavelengths that are absorbed and attenuated due to atmospheric interactions such as Rayleigh scattering.

In a nutshell, the various gases in earth's atmosphere alter the black body spectrum slightly but not insignificantly. In particular, blue light is absorbed and scattered by oxygen gases, and as a result reduces the amount of blue in the resulting daylight spectrum.

Collectively, these adjustments are what we call reconstitution. As a result, the CIE developed a series of D series illuminants to reflect this and more accurately distinguish daylight from black body radiation.

If you take a close look at a chromaticity chart, you will notice that there is a "jump" at 5000K that occurs. This is due to the shift from A illuminant at color temperatures less than 5000K to the D illuminant at color temperatures greater than 5000K. Because the A illuminant is the black body locus, the black body and daylight loci are identical at color temperatures less than 5000K.

One intuitive way to confirm this is to see where the D50 and D65 chromaticity points lie. Indeed, you will find that they are located right on the daylight locus rather than black body locus.

Which should I choose?

First of all, if the color temperature is below 5000K, the loci are the same and therefore there is no need to choose. If the color temperature is above 5000K, whether you use the daylight or black body locus primarily depends on your lighting application and needs. That being said, the daylight spectrum and locus are more commonly used. If you ultimately are looking to mimic natural daylight or any of the D series illuminants, the daylight spectrum and locus would be the way to go.