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The idiom, "Every cloud has a silver lining", is saying that you can always get some benefit from an unpleasant situation. The idiom has a broad application because it does not restrict itself to a particular situation, that is, the idiom can be applied generally.
"With every crisis comes opportunity" is a modern expression. It is also restricted to situations which can be described as a crisis. It is literally closer in meaning to the Chinese expression 有危亦有機.
Which is better? It depends on the message you want to bring home to your readers. In general, a general expression like the first one leaves more room for thought to your readers and a specific one like the second brings home the idea more directly. It is all about the style of writing. Personally, I prefer the first idiom as it is a more beautiful and subtle expression than the second, which sounds opportunist.
Similar expressions:
There are two sides to every coin.
Every rose has its thorns. (The opposite to Every cloud has a silver lining)
2009-04-08 07:02:53 補充:
"With every crisis comes opportunity" is a comparatively modern expression.