Just a few things that spring to mind. There are other similarities and differences of course.

Hardware wallets:

- If you ship the device to your home, they know and store your address. If the data leaks, many more people will know who and where you are.
- You have to trust the developers and manufacturers that they won't mess up and intentionally/unintentionally introduce bugs and vulnerabilities in their native software and hardware. Unless you can go though each line of code yourself (99% of people can't/won't), you will have to trust that the community and security experts have done a good job with it.
- The device (unless airgapped properly) needs to be connected to a computer or phone through a USB cable/Bluetooth to be used. 
- Seeds, private keys, and PIN codes are stored on the device and protected by secure element chips (if available).
- Requires backups of seeds in physical form (highly recommended).
- Change goes to change addresses.
- The coins are spent from the same secure device.

Paper wallets:

- Use TOR or VPN to download the wallet generator to your local machine and no purchase data is stored anywhere.
- You are responsible for your own backups and there are no software or firmware upgrades that can affect you in any way. You are also not affected by NDA agreements or unfixable hardware vulnerabilities.
- No cables, no USBs, or Bluetooth connectivity ever.   
- No digital copies of seeds or private keys. But the same data is imprinted on the paper, making it easier for someone to take note of it.
- Is obviously a backup in physical form.   
- The change could end up in an address you don't have the private key for if you are using such a bad setup.
- The seed/private key has to be imported elsewhere before spending. You could put your coins at risk using a hot wallet for such purposes. Transactions could, however, be created and signed on airgapped systems.