I apologise for a question that has no doubt been covered but searching was not giving me any help.
I know that Japanese cards continued to use the 1st Ed stamp for quite a bit longer than English cards did. However, what does this mean for the rarity of Japanese 1st Ed cards?
Are they still more rare than the same card in English unlimited? By a negligible amount?
From what I understand, some Japanese sets never even had an unlimited print run. Or the 1st edition print ran longer than the unlimited, making some unlimited cards rarer.
I’ve been looking into this recently too. As Galalyth said, different sets had different ratios, so there is no set rule. Someone linked me to this post which was helpful: www.elitefourum.com/t/1st-edition-timeline-lets-iron-it-out/16054/1
From my personal experience looking at Japanese Aquapolis and Skyridge cards it seems that the PSA pop of the unlimited cards are much much lower than the 1st edition cards. Which is apparently true for most, if not all, Japanese E-series sets. But the 1st edition cards still seem to go for money on average. I’m not sure if that’s because it’s not common knowledge yet or because people just prefer having the 1st edition stamp and knowing they’re the older cards.
Anyway I hope that helps, and I have no idea about newer sets I’m afraid.
PSA population does not correlate well with true distribution numbers, in fact you often see the inverse correlation (rarer card = more likely to be graded).
From my own experience, I do tend to see a lot more 1st ed Japanese e-series cards than unlimited. And I believe for expedition, the unlimited is far rarer. But people value the stamp itself and in most cases 1st ed is more desirable than unlimited. The perceived exclusivity of the stamp is usually more valuable than whatever the true relative rarity is.
Well doesn’t that mean the unlimited cards should technically be more likely to be graded, as they’re rarer, which means they’re pop is actually exaggerated and they’re even rarer than we thought? 
No I’m kidding I know what you mean 
But yeah, just from looking to buy cards it’s noticeably easier to find 1st edition cards also.
Expedition was suppose to be 1st Edition only in Japan but some unlimited cards where given out through other means according to this thread that the same person also linked me earlier: www.elitefourum.com/t/japanese-unlimited-e-series-1-cards-base-expansion-pack/16426/1
Thanks for the feedback and! I believe the 1st editions I have are mostly if not all from B&W era sets. I was mostly wondering if it was worth separating them and setting them aside in my binder of “special” cards. In the same vein I have lots of different English promos that I know aren’t worth squat.
Looks like @mimighster already mentioned and linked all the threads I would have linked. Not too surprising, since I’m the guy who helped him recently in this thread. 
As already mentioned, some Japanese UED cards are much rarer than their 1st edition counterpart. Especially for the E series, L2/L3 sets, and the World Championship Pack.
Almost all Japanese sets (ignoring a few minor exceptions) will be harder to find in UED than 1st edition. For their prices there usually isn’t a huge difference, unless the sellers know what they have in their hands. Today I received a lot of about 20 cards with some UED E2 and E3 cards for which I only paid 5 USD including shipping, but I’ve also seen an UED E1 card sell for about 75 USD because the seller (and buyer of course), knew it was a special promo release.
Still looking for two UED World Championship Pack cards to complete my Seviper collection, and also still need the UED E1 Pikachu. Luckily I do have the UED E3 (Skyridge) and World Championship Pack (EX Power Keepers) Pikachus already in my possession. 
Greetz,
Quuador