These words have a shared meaning: "to acknowledge," "to concede," "to make a disclosure." One may admit a mistake or allow that he has made one. Admit also suggests the influence of pressure. In general usage, however, admit and allow are interchangeable in this meaning. The preposition to is unnecessary in such statements as "I admit (to) the mistake" and "I admit (to) having made an error." See also acknowledge.
Dictionary of problem words and expressions. Harry Shaw. 1975.