allow and permit are almost the same, they are used to talk about rules and laws. "Allow" is slightly more common word and "permit" is slightly more formal. "I will not allow/permit this kind of behavior." "Parking your car here is not allowed/permitted." The word "permit" can also be used as a noun, as in "the parking permit", but the pronunciation is different. The noun is pronounced "PER-mit" while the verb is "to per-MIT". "approved" is used for accepting applications or supporting ideas. "The new project was approved by the boss." "The citizens do not approve of the President's actions". "granted" is usually someone higher doing something for a lower person. "The government granted 1 million dollars to the research institute." "God has granted my wish". Out of the four words, this is the most specific word.