What is the difference between a private spend key and mnemonic seed?
Can both be used to sign a transaction? Are there any things that I can derive from one but not the other?
Thanks to deterministic key derivation, the private spend key is an alternate representation of the mnemonic seed. Both the mnemonic seed and private spend key can be used to derive all other keys
The private spend key is used to sign Monero transactions.
Simplewallet mnemonics seed are comprised of 25 words with the last word being used as a checksum. Those words correspond to a 256-bit integer, which is the account's private spend key. With simplewallet the spendkey is the mnemonic seed The private view key is derived by hashing the private spend key with Keccak-256, producing a second 256-bit integer. The corresponding public keys are then derived from the private keys.
MyMonero works in a similar manner but with 13 word mnemonic seeds with the last word being the checksum The 13 words correspond to a 128-bit integer that is used for both spend and view key derivation. With MyMonero the spendkey is the hashed mnemonic seed
Deterministic wallets make wallet recovery and key derivation easy. It is worth noting that you cannot derive the view key from the private key for non deterministic wallets.