You may wanna say what difference between 얘기 and 아기 is.아 is 아기's mainly part.It comes from Chinese phrase "兒".So 아기 stands for babies and childrens.얘기 is the simple spell of "이 애기" which means this babies or this children.If somewhere appear 애기,I think 애기 is wrong expression for 아기,too.
I hope my answer could help you.