The answer is "because I'm being sloppy," but the problem is I don't know exactly where I'm being sloppy. Here's my sloppy argument:
Let $M$ be a smooth compact surface without boundary in $\mathbb{R}^3$ and let $H$ be its mean curvature. If $\langle \cdot, \cdot \rangle$ denotes the $L^2$ inner product, then the Willmore energy can be expressed as $W = \langle H, H \rangle.$ Equivalently, since mean curvature can be expressed as $H = \nabla \cdot N$ where $N$ is the unit normal field, we have $W = \langle \nabla \cdot N, H \rangle.$ But by Stokes' theorem $\langle \nabla \cdot N, H \rangle = -\langle N, \nabla H \rangle.$ And since $\nabla H$ is always tangent to $M$, this inner product vanishes, i.e., the Willmore energy is always zero!
Where did I go wrong? There are several potential flaws -- I suspect that the basic problem is I'm not thinking correctly about how quantities get extended to the ambient space. But I'm having trouble putting my finger on the precise problem.
Thanks!