In both the Ethereum chain specification format and in Parity's documents on private chains, you see the term seal a block. What exactly does that term mean? How is it different than mining a block?

"Seal a block" is a proposed term to describe "mine a block" in a private chain. Private chains typically do not need to use proof-of-work mining, so a different term can be helpful. Specifics of what sealing means is up to a "seal engine", which is a proposed term to describe the consensus algorithm for a blockchain: one could describe that Ethereum currently uses a seal engine of Ethash, and has plans in the future to change the engine to Casper.

Sealing is the generic term for mining for the core devs
I've asked the research slack and this was the answer:

Hi, What does it mean "to seal" a block? is this different than mining? E.g.: "libethashseal: generic wrapper around the POW block seal engine. " http://ethdocs.org/en/latest/ethereum-clients/cpp-ethereum/what-are-the-modules.html?highlight=seal

Eth-Gitter-Bridge @Eth-Gitter-Bridge 13:51 [Christian R] rolandkofler: It is the more generic term

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