Answer
The key difference is the offender. Kidnapping and illegal detention under Articles 267 and 268 of the Revised Penal Code are committed by a private individual who deprives another of liberty. Arbitrary detention under Article 124 is committed by a public officer or employee who detains a person without any legal ground — it is a crime against the fundamental laws of the State, punishing the abuse of official authority.
A related offense is unlawful arrest under Article 269, which punishes anyone who arrests or detains another to deliver him to the authorities without a legal basis for the arrest. So the same act of depriving a person of liberty is classified differently depending on who does it: a private person acting on his own commits kidnapping or illegal detention, while a public officer misusing his position commits arbitrary detention.