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.