Disclaimer: CRI is a broken measure of color rendering accuracy. There are efforts to establish better indices but AFAIK none of them are really widely used. So I'll mention CRI a few times here, but don't take that as an endorsement; datasheets usually show a spectrograph, as well as a CRI value, and if you can relate your needs/wants to the spectrograph instead, you should.
So-called "warm" LEDs (paradoxically, lower colour temperature) tend to achieve their tint by having more phosphor to distribute more of the energy over the longer wavelengths. This results in a smoother spectrum which is probably why they appear to have better colour rendition. It also results in slightly lower efficiency.
There's no such general tendency that I'm aware of. Most neutral- or warm-white LEDs are still single-phosphor, and pick up
at most 5 or so on the CRI scale. They have slightly reduced efficiency, because the absorption/reemission process is inherently lossful, and more of the light is undergoing it. The difference between them and cool-white is, IMHO, insignificant; they render some colors differently, but none of them can really be claimed to render colors
more accurately in general. Not to say "it's all subjective", as certain circumstances may favor particular wavelengths, and make one or the other better.
OTOH, there are 2-phosphor, and I think even some 3- or 4-phosphor, LEDs designed for good color rendering; all the neutral/warm P4s are this way, and some Nichia 083s in cool and warm. They're even less efficient, but move from CRI of 70-80 up to 90+. These are actually substantially more accurate at rendering various colors.
And of course, ability to see through rain, fog, etc. is an entirely different matter from accurate color rendering -- the blue spike in conventional white LEDs causes high levels of backscatter, and while it's dramatically reduced in warmer LEDs, incans are better yet, and a high-CCT incan should school a warm-white LED with somewhat lower CCT (I can't test it myself, as my warmest LED just about matches my Mag61 with full batteries). If penetration through rain is key, an amber LED would be better -- but of course it has much worse color rendition.