Is the Bitcoin Core just the Bitcoin Client and Wallet menagent software or Is it the the Blockchain where the Bitcoin is stored in all the Bitcoin addresses ?
It's wallet management, but also transaction and blockchain management. Bitcoin core is quite a complex software.
However, the addresses are not actually stored anywhere. All the possible private keys are the integer numbers in a huge range. The addresses are derived mathematically from those private keys. So they don't have to be stored.
The blockchain contains
transactions. When a block is mined, a number of transactions are included into the block and the block is added into the blockchain.
When one creates a transaction he needs to know (from the blockchain) exactly what inputs (yes, inputs, not coins) he can spend and will make a transaction off them, will sign it, broadcast it and wait for a miner to mine it.
Blockchain is not a second software; it's just data. The data of all mined transactions until now.