The following is opinion and not fact. It is based on how my own interpretation of plant physiology. I have not had any formal training, just read some textbooks. Mistakes are likely.
It's not so much LED vs. HPS. A photon is a photon no matter its source. They all travel at the speed of light so it's not that HPS has more powerful photons or something like that.
The difference between one photon and another, as far as we know, is the wavelengths.
I believe it has more to do with polymorphism of the plant and how spectrum affects the specific specie. Each species has it own phenotype. As plants evolve the DNA of the cells morph in order to adapt to their environment. With plants, spectrum play a big role in their development environment. Example: how blue, yellow, and far red phytochromes effect elongation i.e. shade avoidance syndrome (SAS).
A good example is a tiger's stripes or jaguars spots are genetic traits. The color and shape are environmental. See a light and a dark-morphed jaguar:
https://en.wikipedia.org/wiki/Morphology_(biology) here in the right sidebar.
Most of the discussions I see on this site regarding spectrum revolve around photosynthesis. It's not about photosynthesis, any spectrum works for photosynthesis. The difference is in how the electrons created during photosynthesis are used. It's about the electron transport chain. Maybe. This is where the environmental decisions are made. It has been found that spectrum has a major influence on polymorphism.
Also consider that a cannabis species grown under HPS has genetically evolved to adapt to HPS and may not do as well under a different spectrum. The DNA morphs will be transferred to the seeds.
Until the legalities regarding cannabis research get ironed out, it may be awhile before we find out. The good news is I was talking to a researcher at a college in Canada (U of Guelph) and he applied for a grant to do some of this research with cannabis.
There is a study done on Sweet Basil. I see some similarities between basil and cannabis, mainly the aroma and oils. This study shows unequivocally how much spectrum affects a plant's development. From this study I am extrapolating if the light recipe can drastically affect the terpenoids and phenylpropanoids for basil, it can do the same for cannabinoids. If you look at teh basil study, just think of cannabinoids in place of terpenoids and phenylpropanoids.
There is a "heatmap" showing how the various light treatment affected the productions of the volatiles terpenoids, phenylpropanoids, fatty acids fatty esters, and etc.
Notice how the column for Blue Red is almost inverted compared to BRGreen. That may explain some of the differences between growing under BR or White. The arguments over white vs. BR the focus often turns to green being the difference. The research I've seen so far on green does not really support that hypothesis either way. I think it is the Yellow spectrum. The basil study supports the yellow hypothesis. It appears to be a very influential color in both the appearance of the leaves and a huge affect on the volatiles.
And while off topic, I may as well add that light also affects the quality of plants (e.g. fruit and vegetables) post harvest. So you may want to store your harvest under various colors. My hypothesis is you can change the flavor just as it has been proven post harvest light treatments can change the flavor of fruits.
The "heatmap" the color treatments are at the bottom of each column. GH is greenhouse.