If the proton were a point charge like the muon, then $ep\to ep$ scattering, the differential scattering crossection is $$\frac{d\sigma}{d\Omega}\Bigg|_{\rm lab}=\Bigg(\frac{\alpha^2}{4E^2\sin^4(\theta/2)}\Bigg)\Bigg\{1+\frac{2E}{m_p}\sin^2\theta\Bigg\}^{-1}\Bigg[\cos^2(\theta/2)-\frac{q^2}{2m^2_p}\sin^2(\theta/2)\Bigg]$$ where $\theta$ is angle scattering, $E,E'$ are the initial and final energies of the scattered electron, $q$ is the momentum transfer in the scattering and $m_p$ is the mass of the proton. In the limit of $m_p\to \infty$, or more accurately, $E,q^2\ll m_p^2,$ the term inside the second bracket becomes unity and second term inside the third bracket vanishes. In this limit, we get $$\frac{d\sigma}{d\Omega}\Bigg|_{\rm lab}\rightarrow\Bigg(\frac{\alpha^2}{4E^2\sin^4(\theta/2)}\Bigg)\cos^2(\theta/2)$$ where the quantity inside the first bracket is the Rutherford scattering formula.
Why does the factor $\cos^2(\theta/2)$ survive in this limit? Why we do not recover the exact Rutherford formula in this limit nonrelativistic limit?