Since this comm is Fandom Grammar and no other, what else can I do but use fannish examples to illustrate my answer? This post’s examples come from the characters of The Professionals.
Let's start with some definitions from The Free Dictionary website. Readers will note that other is a busy word.
2. Additional; more:
1. In a different or additional time, place, or manner:
2. If not; otherwise:
a. Being the remaining one of two or more:
b. Being the remaining ones of several:
2. Different from that or those implied or specified:
3. Of a different character or quality:
4. Of a different time or era either future or past:
5. Additional; extra:
6. Opposite or contrary:
7. Alternate; second:.
8. Of the recent past:
a. The remaining one of two or more:
b. others The remaining ones of several:
a. A different person or thing:
b. An additional person or thing:
1. A different or an additional person or thing:
2. others: People aside from oneself:
In another way; otherwise; differently.
One way that potential usage of else and other is clearly different is that else is used as an adjective or adverb, whereas other can also be used as a noun or pronoun.
Bodie chased full tilt after the two men. One turned left, the other turned right, and he had to make a choice as to whether he followed the one in the denim jacket or the one in the truly execrable jumper.
Clearly in this example, substituting else is going to make a nonsense of your sentence.
At first sight, there is overlap between the adjectival uses of else and other. Generally speaking, other can imply 'not this particular thing but that particular thing', while else tends to imply 'not that particular thing but some altogether different thing'.
Doyle was brooding again, and Bodie wasn't going to put up with it. "You need someone to moan to, you find somebody else, mate, not me."
If you wished to use other in this context, you might have Bodie saying "You find someone other than me." Other often takes than to make its distinctions.
Doyle examined the sandwiches in the cabinet. "These look a bit tired, don't they, love," he asked the dispirited girl behind the counter.
"Well, there isn't anything else," she replied.
To use other in this example, the girl might tell Doyle that "There aren't any other sandwiches."
If it had been any other man than Barry Martin to betray CI5, neither Cowley nor Doyle would have taken it so hard.
Using else you might say "If it had been anyone else to betray CI5...."
Sometimes, other is the only choice for the meaning of your sentence.
Leaping out of bed after getting an emergency call on the RT, Bodie could find one sock but not the other.
The adverbial uses of else and other are quite distinct.
Doyle would recover from his injuries, or else he wouldn't.
"Fine, you hate the job," Bodie said. "Where else would you like to be then?"
The Minister made himself difficult to find, and Cowley was unable to see him other than by pursuing him to his club.
Is there anything else to say? In researching this post, I found plenty of discussion as to the nature and use of adverbs and I include some links on the subject in my list of resources.
Definition of else at The Free Dictionary
Definition of other at The Free Dictionary
Else as defined by the MacMillan Dictionary
Other as defined by the MacMillan Dictionary
Parts of Speech
What Is An Adverb
Determiners and Qualifiers