I suspect there is in general not much difference between GMRES and CG for an SPD matrix.
Let's say we are solving $ Ax = b $ with $ A $ symmetric positive definite and the starting guess $ x_0 = 0 $ and generating iterates with CG and GMRES, call them $ x_k^c $ and $ x_k^g $. Both iterative methods will be building $ x_k $ from the same Krylov space $ K_k = \{ b, Ab, A^2b, \ldots \} $. They will do so in slightly different ways.
CG is characterized by minimizing the error $ e_k^c = x - x_k^c $ in the energy norm induced by $ A $, so that
\begin{equation}
(A e_k^c, e_k^c) = (A (x - x_k^c), x - x_k^c) = \min_{y \in K} (A (x-y), x-y).
\end{equation}
GMRES minimizes instead the residual $ r_k = b - A x^g_k $, and does so in the discrete $ \ell^2 $ norm, so that
\begin{equation}
(r_k, r_k) = (b - A x_k^g, b - A x_k^g) = \min_{y \in K} (b - Ay, b - Ay).
\end{equation}
Now using the error equation $ A e_k = r_k $ we can also write GMRES as minimizing
\begin{equation}
(r_k, r_k) = (A e_k^g, A e_k^g) = (A^2 e_k^g, e_k^g)
\end{equation}
where I want to emphasize that this only holds for an SPD matrix $ A $. Then we have CG minimizing the error with respect to the $ A $ norm and GMRES minimizing the error with respect to the $ A^2 $ norm. If we want them to behave very differently, intuitively we would need an $ A $ such that these two norms are very different. But for SPD $ A $ these norms will behave quite similarly.
To get even more specific, in the first iteration with the Krylov space $ K_1 = \{ b \} $, both CG and GMRES will construct an approximation of the form $ x_1 = \alpha b $. CG will choose
\begin{equation}
\alpha = \frac{ (b,b) }{ (Ab,b) }
\end{equation}
and GMRES will choose
\begin{equation}
\alpha = \frac{ (Ab,b) }{ (A^2b,b) }.
\end{equation}
If $ A $ is diagonal with entries $ (\epsilon,1,1,1,\ldots) $ and $ b = (1,1,0,0,0,\ldots) $ then as $ \epsilon \rightarrow 0 $ the first CG step becomes twice as large as the first GMRES step. Probably you can construct $ A $ and $ b $ so that this factor of two difference continues throughout the iteration, but I doubt it gets any worse than that.