This is not a hard and fast distinction, and in practice there is considerable overlap between the two fields, with methods and results from one being used in the other.
A useful starting point and organizing principle in the study of harmonic functions is a consideration of the symmetries of the Laplace equation.
Although it is not a symmetry in the usual sense of the term, we can start with the observation that the Laplace equation is linear.
This means that the fundamental object of study in potential theory is a linear space of functions.
Second, one can use conformal symmetry to understand such classical tricks and techniques for generating harmonic functions as the Kelvin transform and the method of images.
For this reason, when speaking of potential theory, one focuses attention on theorems which hold in three or more dimensions.
An important topic in potential theory is the study of the local behavior of harmonic functions.
There is Bôcher's theorem, which characterizes the behavior of isolated singularities of positive harmonic functions.
[1] Since the Laplace equation is linear, the set of harmonic functions defined on a given domain is, in fact, a vector space.
By defining suitable norms and/or inner products, one can exhibit sets of harmonic functions which form Hilbert or Banach spaces.