Remaining bitcoins sent to a different address
Your bitcoin client must have used a full UTXO to paid for your transactions, since you didnt own the exact amount your bitcoin client created a new private key where it send the remaining ( 20-5 = 15 ) bitcoins, also you had to pay a fees to the miner who mined the block that got your transaction.

To learn more about this process you should go there https://en.bitcoin.it/wiki/Change 

If you want to receive new coins, The most private and secure way would be to use a new address given by your wallet, I suggest you to learn about HD wallet who can give you automatically a new address each time you want to receive new bitcoins.