Finite differences is the PDE analogue of Runge-Kutta, and it is certainly used (in CFD for example). However, finite element methods have several advantages:
* It can handle PDEs on domains with complicated geometries, while finite differences really prefer rectangular domains. This consideration doesn't apply to ODEs which are always solved on one-dimensional intervals.
* For any numerical approximation it is important to have convergence guarantees, and as the blog post mentions, the analysis is much more well understood for finite elements, particularly on irregular geometries. Strang and Fix's 1973 book is the classic reference here.
* It can handle PDEs on domains with complicated geometries, while finite differences really prefer rectangular domains. This consideration doesn't apply to ODEs which are always solved on one-dimensional intervals.
* For any numerical approximation it is important to have convergence guarantees, and as the blog post mentions, the analysis is much more well understood for finite elements, particularly on irregular geometries. Strang and Fix's 1973 book is the classic reference here.