There may be some vulnerabilities in the hardware or firmware that have not been detected. Also, your hardware wallet can become lost, stolen, broken or corrupted. It is very important to write down your seed words so that you can recoup your coins. The Trezor requires a 24 word seed. Unless you are going to memorize these words in the exact order, you will have to write them down. (electronic storage is not recommended.) However, then that piece of paper can become lost, stolen or destroyed. You will need to safeguard that piece of paper. You can make multiple copies and store them in multiple locations which will mitigate the seed from becoming lost or destroyed. The problem with multiple copies though is that it increases the chance of your seed getting stolen.


Yes, the main reason you may want your private keys is to try and claim some of the hardfork coins. Also, you may want to generate a paper wallet with the private key to use at a Bitcoin ATM and claim cash on the spot. There are ways to claim hardfork coins with the seed. But then you will have to basically send coins to another location and then reset your hardware wallet with a new seed.
The private key is the sting of characters that allows you to spend your coins. It is cryptographically tied to the public key. The public key is the address you use to send coins to.


If you have your seed(you should have written down the 20+ seed words), you can restore the wallet using another wallet program/hardware device. However, you may then want to move all of your coins to another location ASAP. If your hardware wallet is missing, you should assume the seed is compromised. None of the addresses generated by that seed can be considered safe.