Most people mean the same thing when they use them.
What is the different?
Discussion
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.
I guess to put it even more simply, at least for me it comes down to where the key is stored. On device or off device. On device, it's a hardware wallet. Off device, it's just a signing device.