Advantage of having your BTC in a cold wallet
1. You have total control over your funds
2. It can't be stolen or hacked
Example of a cold wallet Is Electrum wallet
Electrum can be used as a cold wallet, however by default or the most common usage it's not. If you ran Electrum in a offline environment, it would be considered cold storage, but even then you have to handle the downloading of Electrum, and the transfer to a offline computer, very carefully. 1. You have total control over your funds
2. It can't be stolen or hacked
Example of a cold wallet Is Electrum wallet
Also, it could technically still be stolen or compromised. There's always a physical threat, and it entirely depends on how you've secured it. Plus, there's no such thing as absolute security, there's always going to be flaws whether that's through the user or the software itself. You can only reasonably secure something, since you aren't going to know all the flaws until they're discovered, and there's been many in the past.
Some don't consider hardware wallets as cold wallets. Then again, they don't consider them as hot wallets either. They are somewhere in the middle. Unless airgapped, they rely on an internet connection for normal functioning, but some are equipped with chips that provide a safe enclosure for your keys and sensitive information.
Technically, they aren't cold wallets. They're pretty much more secure hot wallets in my opinion. Even then, hardware wallets are usually susceptible to some nifty physical attacks. So, again they aren't absolutely secure. However, they're probably my recommendation to new users that new their wallet to be connected to the internet. 
