What's the difference between "another" and "someone else"?

Unless you get into the nitty-gritty, Different and Another can be used interchangeably, but the latter can't take a plural noun i.e, you can't say another people, but you can say different people.

Now, where does the conflict arise?

It's when their attributes such as quality, form, nature, etc come into the equation.

Another replaces someone of similar attributes.

Different replaces someone of dissimilar attributes.

For instance, let's say you're using a Reynolds pen, then if you say, 'Bring another pen.' It means that you're asking for another Reynolds pen. But if you had said, 'Bring a different pen.' It means you were asking for a pen from a different brand that's of different quality or form or nature.

So, if Henry is a plumber, then asking for another man CAN mean you're asking for another plumber.

Similarly, if you have a 1-liter bottle in your hand, and you said, "Give me another,' which means, you want another 1-liter bottle. Had you said, "Give me a different one.' Then, the shopkeeper would have given you one with a different brand, or size, or cost, etc.

Now, coming to Longman's definition: Another means not the same thing, which is right because you're talking about a different person( who can have a similar nature such as profession, etc).

Since different people/things/places have different qualities, forms, and natures, it's best to use someone else instead of another. However, when they have something similar in nature, then you can use another.