tl;dr -- private blockchains make it possible to distribute a common piece of infrastructure between businesses with competing interests. The sit between the stream of new inputs and the database.

I've built one of these types of systems (kadena.io). The key differences between a replicated DB, though let's use consul instead because it's closer to a private/permissioned blockchain (distributed) and the differences are the same, are two-fold:

#1 They are designed to run in a multi-administrative context (i.e. where you don't own all the servers). Imagine running consul where you only had direct access to 20% of the nodes. You could do it, but you'll eventually have problems as it's possible for a few misconfigured servers to slow/halt/collapse the cluster. The problem is that in multi-admin you can't directly fix the troublesome servers. The system itself needs to be robust against them -- you need BFT consensus (for this and other reasons).

#2 The killer feature of private blockchains is having a smart contract language. It allows you to write and run code that runs on my servers without me having to trust/audit your code. This is only possible because of the frankly ridiculous level of security that the system provides. It is also a fundamentally new feature. For a new language that's actually appropriate for serious work (which EVM-based langs are not) take a look at Pact on the aforementioned site.

Simply, if you want to be able to host logic and data, with the infrastructure distributed among multiple administrative units (businesses/firms) that don't completely trust each other, then you end up building a "private blockchain". You need a decentralized way to replicate new information (consensus), that is robust against bad nodes (BFT), that linearizes the new information (blockchain-like), that can execute logic on your machine written by individuals that you don't trust (smart contracts), that tracks/enforces auth at every level (PPK-Sigs everywhere!).