I will take Bitcoin as an example:
Blockchain technology is based on a combination of three concepts that have existed for some time: peer-to-peer technology, cryptography and game theory.
P2P technology: torrents use the p2p technology for distribution of information across the nodes. Nodes on the blockchain network communicate via p2p technology and store information about the blockchain locally. Both concepts (blockchain and torrents) don’t need a central server in order to operate.
Cryptography ensures that transparency and privacy are achieved. Cryptographic hash functions encrypt individual transactions. Blocks and transactions are cryptographically connected, hence keeping the blockchain immutable. Miners check every transaction happening on the network, so cheating is impossible.
Game theory: there is a need of economic incentive in order to participate in the network. Miners would not approve transactions on the blockchain for free. The Bitcoin blockchain pays the miners and this way they stay “motivated”.
So let’s compare both technologies further.
The blockchain is a distributed ledger, spread across the nodes of the network. Its sole purpose is to keep a record of the transactions which are happening. Think of an accountant having their balance spreadsheet opened and checking all the numbers over and over again. Notice, not every node has to store the whole blockchain, mostly miners and wallet providers need to store it locally. Users can just trust the other nodes and send transactions around.
Torrents do not have a distributed ledger to work with, they are only used to spread any kind of information across the network.
Both technologies serve a different purpose. Blockchain technology is all about preventing double spending or in other words preventing people from copying information. Torrents are all about copying information. More copies ensure that you have more seeders. No information gets lost in a blockchain. That is not the case with torrents, where an old torrent could loose all its seeders and disappear from the network.