We study a semidiscrete finite element approximation of the Cahn–Hilliard equation in its symmetric fourth-order formulation, allowing for both constant and concentration-dependent mobilities. Motivated by applications in pointwise control and structure-preserving discretization, we work directly with the single-field problem and avoid introducing the chemical potential as an auxiliary variable at the continuous level. This leads naturally to conforming discretizations in globally C1-finite element spaces, such as the Argyris, Bell, and Bogner–Fox–Schmit elements, which are H2-conforming and well suited for fourth-order operators. For constant mobility, we derive a discrete gradient-flow identity that mirrors the continuous energy law: the discrete Ginzburg–Landau energy is nonincreasing and the dissipation is measured in a discrete metric defined via an inverse discrete Laplacian on the zero-mean subspace. For variable mobility we extend this construction to a weighted discrete inverse Laplacian, yielding an analogous energy dissipation law in a mobility-weighted discrete norm. Using classical approximation theory for globally C1-elements together with relative stability and duality arguments, we obtain optimal a priori error estimates under natural regularity assumptions, and show how the variable-mobility case can be handled via perturbation techniques. Numerical experiments corroborate the predicted convergence rates and illustrate the higher-order accuracy of the Argyris element compared to the Bell element. The semidiscrete analysis is modular and can be combined with energy-stable time integrators.