bones261 pretty much gave a thorough answer so I'll just give you a different answer for the first question.
Personally the only slight disadvantage I have with a hardware wallet is that it's quite a hassle to use compared to software wallets, as you'd have to connect this device to your computer via a usb cable, which is ever so slightly a hassle. No biggie though, having your crypto being secure is a bigger advantage that outweighs this small disadvantage.
This has been the case up until this point (and not to be spammy or anything) but ECOMI are bringing out a hardware wallet (the Secure Wallet) which operates wirelessly using an encrypted bluetooth connection. In this way it is designed to be carried on you at all times, and you can interact with it via the companion app on your iOS or android device, keeping your crypto-assets and private keys as safe as possible, but still have the convenience of keeping it on you at all times.
I have heard they make private key extraction difficult.
2. In what circumstances would you need a private key? What are private keys purpose?
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.
To be fair, the main reason you want your private keys is to protect your assets. The private key is used to encode the information you are sending, and to generate the public key. So if someone has access to the private key they can also access your crypto. Being able to claim new coins in a fork is great, but it can leave you vulnerable to malware and viruses if you're using a hot wallet to claim them. The ECOMI Secure Wallet supports all major forks without ever letting your private keys leave the device, and as it is never directly connected to an online device it can never be corrupted. It will also let you generate QR codes through the app to withdraw cash at bitcoin ATMs without having to carry around a paper wallet.
Point is, cold storage wallets are the safest way to protect your assets, and some are better than others when it comes to security, portability and capabilities. If you want more information send me a message, or check it out at
www.ecomi.com.