Since one is just another representation of the other, is the entropy for the mnemonic seed and the private spend key exactly the same?
Are there any security benefits of using one over the other for wallet storage and recovery purposes?
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The seed is another representation of the private spend key. Because of this, entropy of the seed must equal the entropy of the private spend key.
The benefit of using mnemonic seeds are that you can in theory memorize them and when entering the data, it is easier to avoid making typos when using words instead of long strings of alphanumeric data.