What is the difference between $\dot{x} = Ax + b +Bu$ and $\dot{x} = Ax + Bu$, where b is a constant vector?

If you read the equation as

$$\dot x-Ax=b+Bu,$$

$b$ is nothing but a constant excitation, i.e. a translation of the equilibrium point.

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As you probably know, we can get rid of it with $y:=x+A^{-1}b$, giving

$$\dot y-Ay=Bu.$$