What is the performance difference between Proof of Authority (PoA) and Proof of Work (PoW)? In terms of transactions per second.
Performance: The transaction throughput is limited (among other things) by the block size. By how many transactions one can fit in a block.
Proof of Authority is a centralized consortium chain and can handle potentially hundreds of thousands (or millions) of transactions per second because the scalability is not bounded by consensus on what is the correct or longest chain, but by the hardware used.
This means we can scale vertically, use large data centers to process and store all of this data. We simply allow them to place a large amount of transactions into a single block.
The block producers are few and known. Users of the blockchain trust that they will behave honestly and not collude to perform an attack.
Proof of Work on the other hand does not rely on trusting a small number of block producers to achieve consensus. Instead it allows anyone to participate in block production and achieve consensus on watching what is the longest chain. The transaction throughput on PoW is not (so much) bounded by the protocol, but by a choice users are forced to make between decentralization and transaction throughput. To illustrate this, I will use Bitcoin as an example:
Choice 1 | Small blocks - This means we can only fit a few transactions into a single block which results in low transaction throughput (~7 tx/s). But also means that mining does not require a lot storage. The system is more decentralized.
Choice 2 | Scale Bitcoin through PoA-sized blocks - In this case, users decide that each block may now be as big as 1024Gb instead of Satoshi's 1Mb. Now we can fit many more transactions in a single block and increase bitcoin's throughput. However to run a node you now need high end machines, a lot of storage and a high speed internet connection to process blockchain! Suddenly cannot afford to verify that block producers are following the rules agreed, instead you have to trust them: a regression to the status quo.