Or am I missing something? I had a client ask for White cake on the bottom tier and vanilla cake on the top tier. I've always just used white if they asked for vanilla. I guess I could offer a french vanilla, but I just wanted to make sure I don't tell her the wrong thing like I think I know what I'm talking about!

No there is a difference, mainly the color which comes from use the whole egg vs egg whites

A true whie cake doesn't have egg yolks and uses less vanilla than a "vanilla" cake. If someone described a cake as vanilla, I'd think "yellow" cake with extra vanilla in it.