What significant advantages does BFT provide over On-chain POW solutions and why or why not are they not more commonly used?
Answers can be found in Ethereum's official Q&A on Proof of Stake under the
How does proof of stake fit into traditional Byzantine fault tolerance research? section.
Proof of work algorithms and chain-based proof of stake algorithms choose availability over consistency, but BFT-style consensus algorithms lean more toward consistency; Tendermint chooses consistency explicitly, and Casper uses a hybrid model that prefers availability but provides as much consistency as possible and makes both on-chain applications and clients aware of how strong the consistency guarantee is at any given time.