  We extend the classification of Robert Bryant of Willmore spheres in $S^3$ to
variational branched Willmore spheres $S^3$ and show that they are inverse
stereographic projections of complete minimal surfaces with finite total
curvature in $\mathbb{R}^3$ and vanishing flux. We also obtain a classification
of variational branched Willmore spheres in $S^4$, generalising a theorem of
Seb\'{a}stian Montiel. As a result of our asymptotic analysis at branch points,
we obtain an improved $C^{1,1}$ regularity of the unit normal of variational
branched Willmore surfaces in arbitrary codimension. We also prove that the
width of Willmore sphere min-max procedures in dimension $3$ and $4$, such as
the sphere eversion, is an integer multiple of $4\pi$.
