I've gotten 3 Aesthete stars up to Early Modern and now have to pick my next culture.  Just a bit of thinking out loud while comparing the two.  Grand Secretariat (+1 Influence on Territory) is mediocre.  Though I'm pretty big it doesn't compare to Edo's Shogun's Authority which gives +1 Influence per population.  The difference here is staggering.  

Comparing my current empire...

Ming: 28 territories = +28 influence

Edo: 122 population = +122 influence

However, the emblematic districts may even things up a bit in the short term.  Grand Teahouse = +1 influence per district and +2 influence per adjacent district.  This isn't super clear so I assume that once it is placed in a territory, all of that territory's districts get a +1 bonus.  Edo's Tera is +2 influence and +5 influence per adjacent mountain - a bit trickier placement.

Comparing my current empire...

Ming: 122 districts.  28 territories, let's assume I can get 2 adjacencies in all of them (this accounts for some 3s and some 1s).  122 + 56 = +178 influence

Edo: 28 districts, let's assume I can get a mountain in half of them. 28 + (14*5) = +98 influence

Totals...

Ming: +206 influence

Edo: +220 influence

So they're pretty darn close. However, the deciding factor here comes in the fact that the unique trait starts accumulating IMMEDIATELY, even before I've built any districts, and it lasts every era after.   This seems to make Edo edge out Ming in the short and long run by a magnitude.

Anything else I'm missing here?

Ming is far superior to Edo, +1 influence per district from teahouse can give huge amount of influence. If a city have 50 districts, each teahouse is will give +50 influence from just the district and keep in mind if that city have 5 territories you can get like +250 influence from the teahouse and this number will likely keep increasing as the game progress and the cities get more and more districts.

Oddible wrote:
However, the emblematic districts may even things up a bit in the short term.  Grand Teahouse = +1 influence per district and +2 influence per adjacent district.  This isn't super clear so I assume that once it is placed in a territory, all of that territory's districts get a +1 bonus. 

Per district/territory/pop/tile/specialist EQ are absurdly op compared to other ones.  Because each one counts the whole city, and a good sized city is at least 10 territories.

Yep, anything that says per district counts all districts in all territories within that city. I have had Swedish research labs add around 1000 research per Lab in my biggest city. They scale very, very well.

So a single Teahouse will add a couple of hundred of influence (the extra influence is granted on the emblematic district itself).

I do agree with what everyone has been saying the japanese would only be better if you can take huge advantage of the Mountain bonuses and get like at least 100 influence from just mountain bonuses. Also there are a lot of civics that can give you more influence and with the ming you get them cheaper.