Just wanted to add a little something even though for most people the distinction will never arise.
Sometimes mathematicians define things like a polynomial $P(D)$ in an operator $D$. In most cases the operator $D$ will be a linear operator; which remains consistent with the fact that a linear operator $ T: V \to V$ has a square matrix representation. We know a polynomial in a square matrix is a valid thing and so nothing breaks.
On the other hand, an arbitrary transformation $L : V \to W$ may not have a square representation (say dimensions of $V,W$ are different); so if we just blindly say they are the same thing, one misses this subtlety. So if someone asked me, I would say there is distinction between a linear operator (the domain and co-domain match) a linear transformation (the domain and co-domain need not match) in that every linear operator is a linear transformation, whereas not every linear transformation is a linear operator.