Is 其他人 any different from 别的人? I guess the question could also be, is 其他 any different from 别的?

Is one any more formal, any more slang, any more rude, anything of the sort, than the other?

Or are they essentially the same?
They are similar in the meaning, however, not exactly the same.

其他 has a meaning of "the others", but it also has the meaning of "the rest", since this word is often used in the last when you are listing some categories, for example, "这个项目, 小王负责写代码, 小李负责 code review, 其他人去做测试." this word has the meaning in dividing the scopes.

别的 also has the meaning of "the others", for example, "我只知道这些, 别的我不知道了", in this sentence, the speaker tries to emphasize what he knows ("我只知道这些"), since "别的", from the literature meaning, "别"+"的", 别 has the meaning of different, so this word has the tendency in comparing and emphasis.