What is the difference between linear transformation and linear operator?

For many people, the two terms are identical. However, my personal preference (and one which some other people also adopt) is that a linear operator on $X$ is a linear transformation $X \rightarrow X$. This is why it is common to hear phrases like "Let $T$ be a linear operator on a separable Hilbert space" without specifying the codomain.

They are in a way synonymous, but I learned it this way.

Consider a rotation in $R^2$ of some vector. This is a linear transformation. The operator defining this transformation is an angle rotation.

Consider a dilation of a vector by some factor. That is also a linear transformation. The operator this particular transformation is a scalar multiplication.

The operator is sometimes referred to as what the linear transformation exactly entails. Other than that, it makes no difference really.

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.