The former is for when you control and trust all nodes in your network. The latter is for the more difficult problem of consensus when you don't trust the nodes - otherwise known as the Byzantine Generals problem in distributed systems research.
Only distinction is classical consensus is permissioned whereas blockchains are typically permissionless.
raft and paxos are basically the same, besides leader election, which raft's take makes it simply easier and possible even more efficient[0]. I say "possible" because that depends very much on the consensus state over time, which in most actual workloads can be pretty stable, so at least in some practice, e.g., with hypervisor-cluster like we do, they perform almost the same. The simpler approach of raft can help if you create a library for it from scratch, or for easier understanding when coming into that space, otherwise the differences does not matter too much (in practice), IMO.