Planet Orbit in General Relativity A single planet orbits a sun. With Newton's classical gravity formula, sun moves in a ellipse, with the gravity force always pointing toward the focus/sun. 
With General Relativity model, how does the GR gravity force vector differ from the Newtonian gravity force? Is there a picture somewhere showing the difference? 
With GR model, what is difference between the sun rotating (like our real Sun) and sun fixed?
 A: In general relativity, freely falling particles will move along geodesics, and so in order to find the path $x^\mu(s)$ followed by some particle in a space-time, we have to solve the geodesic equations,
$$\frac{\mathrm d^2x^\mu(s)}{\mathrm ds^2} = - \Gamma^\mu_{\alpha\beta}\frac{\mathrm dx^\alpha(s)}{\mathrm ds}\frac{\mathrm dx^\beta(s)}{\mathrm ds}.$$
To make a connection to Newtonian gravity, depending on the solution to Einstein's field equations, it may be possible to obtain an effective potential from the metric tensor. For Schwarzschild,
$$V_{\mathrm{eff}} = \frac{1}{2r^2}\left(\frac{GM}{c^2}\right)^2\left(1-\frac{2GM}{rc^2} \right) - \frac{GM}{r}.$$
from which you can then find the vector field, as in standard mechanics, $\vec F = -\nabla V$. As for your other question, the difference between a stationary sun and a rotating sun is that the former is then described by the Schwarzschild metric, whilst the other, by the Kerr metric.
Returning the concept of an effective potential, rather than the one above, we might want to approximate a rotating spherical body with,
$$V_{\mathrm{eff}} = \frac{1}{(a^2+r^2)^2-a^2\Delta} \left\{ aL(a^2+r^2-\Delta) + \sqrt{r^2\Delta \left[L^2r^2+M^2\left((a^2+r^2)^2-a^2\Delta)\right) \right]}\right\}$$
where $a = J/Mc$ and $\Delta = r^2 - 2GMr/c^2 + a^2$. As you can see adding in rotation significantly complicates the effective potential. For its orbits, see this paper.

There's a lot to read in the paper, but this is a plot of $V_{\mathrm{eff}}$ for varying radius $r$ and $L$. The unstable and stable orbits correspond to maxima and minima. Circulator orbits are solid curves.
