The wallet is a container for private keys. BTC wallet cannot directly store Bitcoin, so if you lose your private keys, you will no longer be in control of the assets. BTC wallets contain private keys and corresponding public keys. You are in sole control of the private keys and can use them to generate digital signatures to execute trading.
Ethereum is a decentralized software platform that enables smart contracts and distributed applications (DApps). Ethereum is not just a platform but also a programming language that runs on a blockchain. This helps developers to build and publish distributed applications. ETH（Ether） is a necessary element — a fuel — for operating the distributed application platform Ethereum. It is a form of payment made by the clients of the platform to the machines executing the requested operations.
Ethereum was proposed in late 2013 by Vitalik Buterin, financed by ICO in 2014. It attempts to become the next generation of decentralized software platform that enables smart contracts and distributed applications (DApps).