this is a difficult distinction for sure! nice question! so they are both indefinite pronouns, so that means that we are using them to replace a noun, but since they are indefinite the noun that they are replacing has probably already been mentioned or it's not that important for the meaning. So the main difference between "other" and "another" is that "ANOTHER" is referring to some additional person or object (similar to or different than the one we already have)
1. That was a really good lemonade, i would like to have another (so that's an additional what? lemonade! I'm using "another" as a pronoun for lemonade and because it's "another" that means it's an additional one on top of the one i already have)
2. That movie was good, i would like to see another! (so that's an additional movie! In this case i mean a different story, but still a movie)
Now "other" also means an additional thing or person but except the one you already had or mentioned.
1. that was a really good lemonade, but now i want some other drink (i'm saying that i do want another drink but not a lemonade, i want something different now)
2. This movie is not good, let's go to this other one instead (I still want to see a movie, but definitely not this one that we're seeing now)