I have bright caps only, but plan on getting flat caps.
Warning, subjective assessment follows..
An obvious point of comparison is the DPA 406x mini omnis, with which I am quite familiar. I've run comparisons between B3 (+bright cap) and 4060(+soft boost grid). The specifications for the B3 bright cap and 406x soft boost grid show approximately the same +3dB response in the treble, and since I almost always use the soft boost grids on the DPAs I figured that comparison is appropriate.
To me, the 4060+soft grid sounds like it has a bit more of a broad, high shelf boost compared to the B3+bright cap, so it sound somewhat brighter and more extended overall. The B3 sounds like it has less of that broad boost, but more of a peaking resonance somewhere in that region. Some have mentioned that they hear a fatiguing high frequency resonance with the DPAs. Subjectively, I hear that with the B3+bright cap much more than the 4060+soft boost. I assume that those who prefer the B3 in that case are probably using the flat cap which may not produce that resonance or whatever it is, or that their hearing perception differs significantly from my own.
I plan to do additional comparisons once I get some flat caps for the B3s. To be clear about what I listen for and how I use these, I do not expect the B3+flat cap to have the same response as the 4060+soft boost. Instead I will be listening for the absence of that resonance or whatever it is, with the expectation that I will then be able to EQ the resulting frequency response to my preference. My ultimate test for mic preference is that which provides me the cleanest recording which I can mold as required. If I can't easily manipulate the resulting recording into what I want, it doesn't matter to me if that mic's raw frequency response is slightly closer to the desired target than another mic which does allows me the freedom to dial things in perfectly. That's my way of working and assessing these things. Others have different but no less valid priorities, such as which mic's raw response sounds closer to what they want, for the way they record most material, without any other editing.
I'm with Brian on generally wanting a flatter response up close and a slight diffuse type boost farther away. I'm willing to do that tailoring with EQ if it works better for me than changing caps/grids, so I'm really looking for the cap that does the least harm. Like he says, best way to find out is to try them yourself.
Since I never use the high boost grid on the 4060 (far too peaky) I imagine would not use the very bright cap on the B3 either.