Compare to the commonly used ERC 20 standard, ERC223 is backwards compatible with it meaning every software that supports ERC20 will also supports ERC223.
The biggest change is that ERC223 no longer allow token to be transferred to a contract that does not allow token to be withdrawn meaning if someone send their token into such contract that has not allowed anyone to use it, the token will simply be locked and can never leave that contract. It can also save transaction fees in terms of gas.