Thayer, as opposed to Trench and others, would make the distinction between these two words to be this: "αἰτέω signifies to ask for something to be given, not done, giving prominence to the thing asked for rather than the person, and hence is rarely used in exhortation. ἐρωτάω, on the other hand, is to request a person to do (rarely to give) something; referring more directly to the person, it is naturally used in exhortation, etc."