First of all, I read this answer and I understand that Weyl transformations are transformations of the metric and Conformal transformations are transformations of the coordinates that "Weyl-transform" the metric, but from what I see Weyl and Conformal invariance are pretty much the same thing: doing something that only changes the metric by a positive rescaling.
In my String Theory course, we talked about the $d=2$ worldsheet of the string being Weyl-invariant, and how this invariance (particularly in $d=2$) is important. Later on, we moved to AdS/CFT, and studying the properties of a CFT we mentioned that it is conformally invariant (duh) but since it generally is in $d>2$ it won't be "as nice as" the $d=2$ invariance of String Theory. Why is that? What changed?