Sorry for the long title, but let's make it immediately clear: the keys used to sign transactions never leave the card and cannot be exported. You can however export any public key as well as the private key of keypaths defined in the EIP-1581 specifications. Those keys, by design, are not to be used for transactions but are instead usable for operations with lower security concerns where caching or storing the key outside the card might be beneficial from an UX point of view. Of course, exporting a key always requires user authentication.
The export command is very powerful, since it allows you to derive & export a key in one step. You have the option to make the derived and exported key active or leave the active key untouched. You can also decide whether to export only the public key or the entire keypair (following the rules defined above).
A very convenient use case is deriving an account key and retrieving the public key in one step. This is faster than doing it with two commands (derive key and export public), because every command processed has some overhead. Example
Another use case, is to export keys defined by EIP-1581 without changing the current active key, since you won't be signing with the exported key using the card
For non EIP-1581 paths you can also export the extended public key, including the chain code. This allows, for non-hardened keys, to derive child public keys.