The base proof of work system only differs on the block time, and the hashing algorithm. Bitcoin uses `sha256d`, ethereum uses `ethash`.

Ethereum currently does not make use of Proof of Stake, although it is planned for the future. Some of the testnets use Proof of Authority.

The consensus rules differ by a fair extent, however, since the two chains follow different models. Bitcoin is a utxo based system, and the consensus rules revolve around that. Ethereum is an account based system, and has its own consensus rules to deal with that.

Moreover, ethereum has a much more complete smart contract platform, which requires additional rules, such as ensure transactions are executed in order.