The difference is in population sizes. With (just) bremsstrahlung, you're talking about a single electron passing near an atom. This results in the typical expression of the radiated power,
with $\gamma$ the Lorentz factor and $\boldsymbol\beta=\mathbf v/c$. This leads to $P\propto a^2$ (with $a=\dot\beta$), so the emission depends on the deceleration of the electron as it passes the atom.
With thermal bremsstrahlung, you're talking about a (large) population of electrons that follow the Maxwell-Boltzmann distribution (a thermal population since the electron distribution depends on $T$) and are undergoing the bremsstrahlung process. This leads to the power of
so the emission here depends on the temperature of the gas cloud.
For more information, you can take a look at Chapter 5 of Rybicki & Lightman (Amazon link), which discusses single electron, thermal & relativistic bremsstrahlung emissions processes.