At least for me, the difference is a hardware wallet stores your keys on the device. That would be something like a jade or a passport.

A signing device is just that, a signing device. It doesn't store your keys long term. You have to load your keys on it to be able to use it to sign your transactions. Like a seed signer.

A hardware wallet can sign your transactions for you because it stores your keys on the device, but a signing device is only used to sign transactions.