In this work, a comparative study for two simulation methods is conducted for interfacial flows in two dimensions: an arbitrary Lagrangian Eulerian (ALE) finite element method (FEM) on interface-conforming meshes and a phase field lattice Boltzmann method (LBM) on Cartesian meshes with quadtree adaptive mesh refinement (AMR). The methods are validated by simulations of a bubble without and with buoyancy force. In particular, a suspended bubble with initial nonequilibrium shape and a rising bubble driven by buoyancy force are simulated to validate the methods. Additional simulations of the breakup of a rising bubble and the bubble interaction with a horizontal wall are used to quantify the efficacy and efficiency of the two methods. It is observed that the phase field LBM is more dissipative due to the nature of the diffuse interface method used to capture the interfaces. Overall, the results obtained from both methods agree well with each other when the effects due to the numerical artifacts intrinsic to the diffuse interface method can be neglected. Also, the LBM is in general more efficient and easier to be parallelized.