A fair bunch of Chinese characters are also actually pictographic (they're an image of something) or ideographic (they represent an abstract idea, not a morpheme or word) in origin, if not current usage.
This stuff - synergies and conflicts between writing systems and various media - fascinates me. Chinese writing for example now suffers from the problem that if your medium doesn't allow free-form painting/graphical compositing (like our computers largely don't right now) you can't easily coin new logograms, since you need a central registry that's slow to distribute to leaf nodes (cf. Unicode). So what's happening now is that new words get coined by recombining existing characters based on their sound value, adding a phonographic layer on top. And of course inputting Chinese has also become dependent on auxiliary systems like pinyin-based IMEs.
Meanwhile, the Korean Hangul alphabet has the interesting property that some of the letters graphically derive from each other with consistent patterns of "take this, add a stroke and you get this", making it easier to design keyboards with reduced numbers of buttons to enter combos. There's the theory that this contributed to the success of texting/mobile in Korea, and is a contributing reason for the healthy mobile industry there.
I'm getting far from the original topic with this rambling, but it still reads on it in one sense: There's a lot of variety to writing systems, and it's worth thinking about emoji as a spot on the spectrum instead of in isolation.
Except that emoji aren't a writing system. Writing systems encode language. Emoji encode emotions, perhaps, or suggest collections of ideas/emotions, but don't convey language per se because there's no conventionally defined correspondence between words in a language and a given emoji.
If I give you a string in English or Chinese or Arabic or Amharic, using the conventional writing systems for those languages, each string will be interoperable as a specific sequence of words in that language. A string of emoji doesn't work that way.
Edit: to circle back, you said:
How are emoji different from other logograms?
and everything thereafter was pointless pedantry on my part. Sorry for wasting valuable minutes of your life.