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I wonder if these two sentences mean exactly the same, or is there any difference?

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    Sense is the same, but the former has a negative word which the latter doesn't have. – Ram Pillai Dec 20 '19 at 19:40
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Logically, they mean exactly the same thing.

But in practice, thy have a slight difference.

If someone is being negative, feeling that they can't accomplish what they want, you would respond with something that matches their mood: "nothing is impossible". You deny their negative attitude with another negative.

But if someone is being hopeful about what they want, but with doubts, you would respond with something that matches their mood: "anything is possible". You encourage their positive attitude with a positive statement.

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They mean the same thing! There's no difference besides the use of different words.

Venn Diagram

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    A Venn diagram would help support this answer. – AmI Jul 3 '18 at 19:34
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    Added! Thanks for the suggestion on how to strengthen my answer! @AmI – jmrpink Jul 3 '18 at 21:05
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If you consider one thing can either be possible or not (so symmetry among sets without intersection), you should have the two circles that do not intersect. This implies that "nothing is impossible" differs from "anything is possible"

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