Bitcoin core: It is an old brand Bitcoin wallet with full node. There are Window and Mac versions. Its advantage is complete functions. We can not only check blockchain information, trading records, addresses, public keys and private keys, but also can make encryption for the wallet under command lines mode. However, after the wallet is downloaded and installed, it will be automatic synchronized to network which will takes many days to download all blockchain -150Gb. Therefore, there will be high requirement for hard-disk space.
I know this one its really good, but the size is too big for some people if they just want to store their bitcoin. For alternative i recommend you to use
Electrum. Its almost same as bitcoin core.
Blockchain: It is a special mobile light Bitcoin wallet which supports storage of Bitcoin and Ethereum(ETH). As it is a light wallet, the storage of whole app and its data is not more than 50M.
AFAIK, you need register via dekstop/website version first. After that scan QR code on it to login through your phone. If you go to
Web Wallets section, you will find some problem on blockchain.info wallet.