To simplify hugely:
CS25 - the aircraft should never be endangered by an engine failure during takeoff. At all points it can either stop safely or continue the takeoff.
CS23 - Until the aircraft is clean, there is no guarantee of being able to either stop or climb away in the event of an engine failure. CS23 multi engine aircraft must show some very minimal OEI clean performance, single engine obviously not!
There are CS23 aircraft that are capable of meeting the CS25 requirements, and the manufacturer may or may not give you the graphs to be able to operate it that way, but the underlying principle remains that a CS23 aircraft that loses an engine is quite likely to be in a runway overrun or forced landing situation. There are quite a few pilots flying aound who don't realise this, and think that having a second engine gives them the same safety level as a B777.