There is one significant difference between hot and cold wallets. A hot wallet lives on a device that can be connected to the internet, while a cold wallet stays offline. The cold wallet's isolation keeps your private keys away from malicious attackers, which in turn, improves the overall security of any cryptocurrency you own.
But before getting too deep into why hot and cold wallets matter, it's worth exploring what a cryptocurrency wallet does. Despite their name, cryptocurrency wallets don't actually hold any of your funds within them. Instead, it's best to think of any cryptocurrency you own as a balance recorded on a blockchain network, like Bitcoin or Ethereum. Your wallet simply holds the cryptographic keys necessary to prove ownership over your balance, similar to an online bank account you'd sign into to manage your money.
A paper wallet is the most basic kind of cryptocurrency, as it's just a physical piece of paper. It includes the private keys that allow you to access your cryptocurrency on the blockchain, commonly represented as a list of 12 or 24 words. However, you must still use an app or program to authorize transactions through a process called signing.
Many modern wallets are primarily designed around convenience. Software wallets like smartphone or desktop apps offer the most assistance, allowing you to sign transactions from anywhere. Hardware wallets, on the other hand, keep your private keys isolated on a dedicated device. You'll still use an app to initiate transactions, but the signing process takes place entirely on the dedicated hardware. This is why you typically need to connect a hardware wallet to your phone or computer via USB, Bluetooth or NFC.