According to this HiNative question, ～みたいだ tends to be used subjectively, whereas ～らしい tends to be used more objectively.
This seems to indicate that the difference is the same as that between the 2 senses of ～そうです, i.e. deduction vs hearsay.
Is the distinction as simple as this?
Yes. みたい is usually about your perception or opinion. らしい conveys indirect information that you heard, saw or read somewhere.
Used when you think (subjectively) something looks or is like so. Used very often.
お母{かあ}さんは怒{おこ}っているみたいだ。
彼{かれ}のことを好{す}きになってしまったみたいだ。
そのチケットは売{う}り切{き}れてしまったみたいだ。
Used when reporting that something looks or seems like so according to what someone else said, or something you read. It's more objective than みたい.
お母{かあ}さんは怒{おこ}っているらしい、弟{おとうと}がそう言{い}っていた。
イタリアまで飛行機{ひこうき}で15時間{じかん}かかるらしい。
新{あたら}しいお店{みせ}が来月{らいげつ}オープンするらしい。
Another interesting use of らしい is to express that something is the archetypal rendering of a definition or concept. You use it when someone or something looks like the best representation of what is expected.
In other words, that something is just what it should or ought to be (according to convention / tradition / common knowledge, etc). For example, 男{おとこ}らしい means masculine / manly and carries a positive connotation, whereas 男っぽい is just mannish or kinda like man, also it isn't clear whether it's meant as a compliment or insult. Another example is 子供{こども}っぽい childish.
This is a good example:
お兄{にい}さんらしくしなさい。
Act more like a good older brother! or That's not how a good brother should behave!
