A new stable continuous-in-time semi-discrete parametric finite element method for Willmore flow is introduced. The approach allows for spontaneous curvature and area difference elasticity (ADE) effects, which are important for many applications, in particular, in the context of membranes. The method extends ideas from Dziuk and the present authors to obtain an approximation that allows for a tangential
redistribution of mesh points, which typically leads to better mesh properties. Moreover, we consider volume and surface area preserving variants of these schemes
and, in particular, we obtain stable approximations of Helfrich flow. We also discuss fully discrete variants and present several numerical computations.