They are different methods and presence of a Riemann solver is the distinguishing characteristic.
Finite Difference WENO starts with vertex values, evaluates the fluxes associated with left- and right-traveling waves, and uses weighted averaging to define a flux at staggered points. (In the paper, the average is done in local characteristic variables.) FD WENO is inexpensive, but only works on smooth meshes and requires a flux splitting.
Finite Volume WENO reconstructs a piecewise discontinuous state and evaluates it at quadrature points on cell interfaces. A single-valued flux is evaluated using the multivalued state at those quadrature points. FV WENO is more general, but Riemann solvers are often more expensive and most full-accuracy variants end up needing many more evaluations in multiple directions (due to the need for transverse reconstruction). See Zhang, Zhang, and Shu, On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes, 2011 for more details.