The jump between Ethereum to EOS isn't nearly as big as was from Bitcoin to Ethereum. The same concept of using smart contracts applies, but EOS has the following "improvements":
1) Ethereum is currently partially proof-of-stake but will be fully soon. EOS is launched as completely proof-of-stake.
2) EOS has no transaction fees. The developers realized that fees limit the growth of the network, because no dAPP running hundreds of transactions per second wants to pay a fee. The incentive to "Stake" your EOS tokens and confirm transactions as a block producer comes from inflation... at the rate of 5% per year.
3) EOS will certainly handle thousands of transactions per second when they mainnet launch in June. Ethereum currently has 7 tx/sec, which will very likely increase significantly. However, by that time EOS could potentially be running hundreds of thousands of tx/sec as their code is meant to run on parallelized servers.
4) Uses 21 block producers that act like state governments to that can vote to make changes to the constitution. Users and dAPPs vote for block producers that they think best benefit the community. Ethereum dAPPs don't have this governance structure, but there are a few emerging (ie. 0x).
5) Plans to implement cross-chain communication using state channels. This would mean EOS interoperability with Ethereum and Bitcoin. Technically Ethereum can do this, but EOS is heavily developing on this concept at the moment.