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 ensuring transactions are executed in order.