We use finite element methods (FEM) because of a combination factors:

• To solve problems on complex domains as well as to enable mesh adaptation, the meshes we use to discretize the domain are easier to construct when they are composed of simplices (triangles in two dimensions, tetrahedra in three dimensions, and pentatopes in four dimensions).  Finite element methods work naturally on simplex meshes (in fact, any mesh type).  
• Further, higher-order FEM are easy to derive simply by increasing the polynomial space used to construct the elemental solution representation.  And, higher-order FEM maintain the same elemental computational stencil as their lower-order counterparts.  In contrast, the typical path to higher-order in finite volume methods involves extending the stencil.  This extended stencil approach causes difficulties on unstructured meshes, often leads to a lack of robustness, and requires modifications near domain boundaries.
• The theoretical foundations of finite element methods, in particular the weighted residual formulation, provide a direct route to error estimation which can then be used to drive adaptation.