Hi Alex..Welcome to DC. Come in a make and yourself at home.
I'm not a baker, but quite possibly the difference is the flour. Be patient and someone will come along to give you the details on the different flours...
Also, most "grocery store" cakes are not baked on site, but rather come in frozen by the case and are thawed and decorated as needed. Oh, in very broad terms the icing they use is mostly shortning & sugar based.