These two words appear to mean the same thing and the entry on Google dictionary gives the definition as the same for both.
Are there any cases where you would use one instead of the other?
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Sign up to join this communityAlthough it's etymologically a compound of 落{お}ち+入{い}る, it's now usually written 陥る instead. The NHK漢字表記辞典 recommends writing it 陥る and doesn't mention the other spelling at all. Some dictionaries list both spellings, as you point out; for example, 明鏡国語辞典 lists the word under 陥る but mentions the alternative etymological spelling:
〔表記〕語源を反映させて「落ち入る」とも。
This is written under sense one, 「落ちて、中のほうに入る」, so I suppose this spelling would likely only be used when the word is used in its literal sense, as reflected by the kanji 落 and 入. But if I try to check actual usage by searching the Balanced Corpus of Contemporary Written Japanese (BCCWJ) using the freely available 少納言 interface, I find the following results:
陥る 600 results
落ち入る only 1 result
So regardless of meaning, writing it as 落ち入る is a lot less common. I think you can stick to writing 陥る yourself, and be prepared to recognize the alternate spelling if you ever see it.
It's all about how ancient Japanese decided to make correspondences between native Japanese words and Chinese characters. A native Japanese word おちいる is undoubtedly composed of two native Japanese words おち[る] and いる. But one ancient Japanese guy found this particular combination would better translate to a single Chinese character, rather than a combination of literal translation of each of them
When I write おちいる in 漢字、I always write 陥る. I don't think I've ever seen the case of おちいる being described as 落ち入る. MS Word simply converts 'ochiiru' into 陥る, with no alternative.
We hear 落ち込む meaning "depressed" pretty often, but no 落ち入る.