Can't say I've had much experience with BD1N and what I have had is unfavorable, to me. To preface, I'd like to first and foremost think the appreciation of BD1N will be maximized and truly seen within a kitchen setting, versus a folding pocket knife.
It is a very fine-grained steel, with little alloy in its matrix. The extra carbon, compared to 8cr will allow it to get harder in heat-treatment, so theoretically you will achieve higher attainable sharpness and with much ease (IME). I've heard no real problems with anyone getting 8cr very sharp, aside perhaps the gummy feel on the stones and potential issues with burr/burr removal; YMMV. The addition of Nitrogen also allows much resistance to corrosion, a notable increase when compared to 8cr. Likely the biggest discernable difference to users between these two will be in sharpening response and corrosion resistance.
VG-10 is a class above BD1N. No real comparisons there in use; as I find VG-10 to sharpen at least with equal ease and in every other discernable category, in my experience, exceeds BD1N. This is strictly speaking in pocket knife terms and specifically, Spyderco. Corrosion resistance could (and will) be disputable. One difference I can note is BD1N will roll and VG-10 will chip, if stressed in a similar manner, other attributes being more-or-less equal (hardness, geometry, use...).
While I have no experience with 8cr, I can estimate that at least looking at it on paper and given Spydercos renowned investment in their heat-treating, BD1N will certainly be more similar than different in terms of experience, within a pocket knife and for its uses. My best comparison would be to AUS-8 and AUS-8 itself gets a rep for being worse than 8cr in performance, from many folks. Take that for what you will. Edge longevity in this class of steels is trivial and has more to do with your geometry and edge finish choice, versus anything else, IMO. The ease of achieving and maintaining that edge with minimal effort and resources is the main draw to this class of steels.
All this being said, overall BD1N is not a bad steel, but it doesn't interest me in any more of a way than many other "lesser" stainless steels do. Pocketbook and curiosity subsiding, there are better steels for me, for my money. If you value ease of sharpening and the ability to achieve and maintain a high level of sharpness, along with the added corrosion resistance and things like trivial differences of simply having a harder knife, by all means BD1N may be the steel for you. Many factors at play in making a decision, besides blade steel alone.
In ending, I will state, scientific testing will tell you one thing and experience will tell you another, in regards to many of the differences that you will notice in having one blade steel vs. another. Sharpening ability and resources available would be my biggest concern, when it comes to steel and performance.
Hope this helps.