Thanks for the link.
Typically, Neutral White emitters are one (or two) output bins lower than concurrently shipping Cool White versions. This is due to the extra phosphor required to "warm" up the color temperature (i.e., net effect is to reduce luminus flux, given common manufacturing processes). So that would mean anywhere from ~1-13% lower output (~7% lower on average) for any one bit-step down, depending on where exactly the two samples are within their respective bins.
But you can find Neutral White U2-bins. Although uncommon, there would be no expected meaningful difference in output to a Cool White U2-bin emitter. Of course, they could vary up to ~7% (i.e., depending on where each one is within than common bin). And there are a lot of other variables between actual light samples (i.e. circuit, reflector, etc.) that can modulate that further. But the point is that the output bin is what matters in terms of output - regardless of the tint.
In this case, I see there is at least one other vendor that is reporting NW T6 output bins for the TN12-2014. That would be more in line with expectations (i.e., one output bin down).
You can't discriminate one output bin difference in real life (especially when tint changes), so the only real decision question should be which tint you prefer.
P.S.: As an aside, try not to ascribe too much meaning to tint comparison beamshots. It's not really possible to find a single common camera white balance that will match your relative perceptions.