What to consider between Compton scattering and Thomson scattering? Both Compton scattering and Thomson scattering are scattering of the photon from free electrons. How to decide in a given situation whether I need to use the Compton formula or the Thomson formula?
 A: You can decide this depending on the energy of the incident photons.
When the energy of a photon is much less than the rest mass energy of the electron
($h\nu \ll m_e c^2 = 511\text{ keV}$, i.e. for infrared, visible, ultraviolet light) 
then you can safely neglect relativistic effects.
That means, you get a neglectable wavelength shift of the photon,
a neglectable recoil of the electron,
and you can use the formula of Thomson scattering as a good approximation.
For not so low-energy photons (i.e. for X-rays, gamma-rays)
the relativistic effects are not neglectable.
That means, you get a wavelength shift of the photon, a recoiling electron,
and you need to use the formula of Compton scattering.

Let's elaborate on the formulas for the cross-sections.
The differential cross-section of Compton scattering is given
by the Klein-Nishina formula
$$\frac{d\sigma}{d\Omega}=
 \frac{1}{2}r_e^2\left(\frac{\lambda}{\lambda'}\right)^2
 \left(\frac{\lambda}{\lambda'}+\frac{\lambda'}{\lambda}-\sin^2\theta\right)$$
where $\lambda$ is the wavelength of the incoming photon,
$\lambda'=\lambda+\frac{h}{m_e c}(1-\cos\theta)$ is the
wavelength of the outgoing photon,
and $r_e=\frac{e^2}{4\pi\varepsilon_0m_ec^2}$ is the so-called classical electron radius.
For low-energy photons ($h\nu \ll m_e c^2$, or equivalently
$\lambda \gg \frac{h}{m_e c}$)
you get $\frac{\lambda'}{\lambda}\approx 1$, and the formula above reduces to
$$\begin{align}
\frac{d\sigma}{d\Omega}
\approx \frac{1}{2}r_e^2 (2-\sin^2\theta) \\
= \frac{1}{2}r_e^2 (1+\cos^2\theta)
\end{align}$$
This is the well-known differential cross-section of Thomson scattering.
A: It depends on whether you want a relativistic solution or a classical solution. Compton scattering is just the quantum relativistic version of Thomson scattering.
One should generally use Compton scattering because it gives the correct solution. Thomson scattering cannot explain the shift of wavelength of light in small intensities. One should use Compton scattering in low intensity cases. In high intensity cases the values for cross section of Compton scattering and Thomson scattering should be the same.
