E.G. お忘れなき ＆ 迷いなし

I know they're both ways to end a a sentence, but I never could figure what
make each different from the other. Anyone have a clue?
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grammar
What separates なき from なし？
なき can never be used to end a sentence. In Classical Japanese, なき is the 連体形
of なし, which basically means it modifies nouns:

> 海なきところ - a place without an ocean

On the other hand, なし is what you use at the end of a sentence:

> そのところは海なし - that place doesn't have an ocean

In the Muromachi period, なき was indeed used at the end of sentences, but that
was Late Middle Japanese, not Early Middle Japanese, from which Classical
Japanese was derived.

(In general this applies to all adjectives. Use し (rather than modern い) at
end of sentence, き when modifying. 古き日本語は古し XD)
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The answer in terms of classical Japanese is probably what you're looking for,
as the modern usage will be -based- on its proper classical usage, but
basically なき is 連体形 and なし is 終止形. In modern Japanese the distinction between
連体形 and 終止形 has deteriorated, but if I were to stress a distinction, 連体形
functions in modern Japanese (when it is -not- attaching to a noun) as if it
has an explanatory/declarative の/もの/こと at the end. In modern Japanese, that
"explanatory の particle" you learn is just a marker to make clear the
preceding verb is in its 連体形 form (though this の is not always necessary).

This is -partially- due to the remnants of the 係り結び bound-particles system,
and is also similar to why sentences can end in both か and のか (though they
both have to be in the 連体形 technically), which started degrading even before
we really hit modern Japanese. One other way of putting it is that -because-
the ending is 連体形 and not 終止形 it implies that something that is supposed to
follow was omitted.

お忘れなき on its own is basically: お忘れ（は）ないこと. I'd say it's closest to
"お忘れ（し・は）ないようにね"

迷いなし is simply a statement. 迷い(は)ない. Depending on context this could be "I am
not lost/hesitating etc." basically to mean "All's fine here."

Hope that helps.

--

Since we were talking about it in the comments I'll add another usage of なし.
終止形 can be used for nominalization and なし is a common case that forms (adj-no)
"adjective-type noun" when used for nominalization. In this case it acts like
an na-adjective but uses の instead of な (because it is technically a noun, not
an adjective).

A very common example is the word: なけなし. "なけなしのお金" - a small amount ("not no")
of money.

Other examples given by snailboat include: 「ネクタイなしの軽装」「文句なしだ」「アスプリン・砂糖なしの錠剤」

If I were to say there were a difference between 文句なし and 文句なしだ... The former
is simply "there are no complaints" and the latter is "(in describing the
situation you have referred to/we are talking about) there are no complaints".
An insignificant nuance by itself, but using the だ would be more appropriate
when tying it in with a larger sentence structure. "前のことについてなんですけど、それは文句なしだ。"

--

The difference between べき・べし works similarly, just with more irregularities,
because べき itself is treated as a na-adj nowadays.
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