Often when numerically solving nonlinear PDE's using method of lines approach with an implicit integrator a system of nonlinear equations have to be solved. To be more specific, let's say we have simple backward Euler method: \begin{align} y^{n}=y^{n+1}-h f(y^{n+1},t)= G(y^{n+1},t) \end{align} From the available literature, it seems the most common approach is using an Newton-GMRES method to solve the nonlinear system, especially if it is stiff. As far as I can understand GMRES is used because it is a matrix-free method and so also works with vector functions which is good when approximating the Jacobian with finite differences.
My question is then, why not just use the GMRES to solve the initial problem? I see no reason why GMRES couldn't solve this for $y^{n+1}$ when for the Newton-GMRES we solve the system: \begin{align} H(y^{n+1}_k) &= y^{n+1}_k-y^{n}-h f(y^{n+1}_k,t)\rightarrow 0 \text{ for } k\rightarrow \infty\\ Jv &\simeq \frac{H(y^{n+1}_k+\epsilon v)-H(y^{n+1}_k))}{\epsilon} \simeq -H(y^{n+1}_k)\\ y^{n+1}_{k+1}&=y^{n+1}_k+v \end{align} where the GMRES is used for the second part to find $v$. I don't see why solving for $v$ with GMRES followed by the Newton step should be easier/better than solving the general system in the top for $y^{n+1}$ using GMRES? Does the finite difference approximation to the Jacobian have better properties compared the the general system $y^n=G(y^{n+1},t)$?