Take for example the well known
$$ \ddot{\phi} + \frac{g}{l} \sin{\phi} = 0.$$
Usually, in a physics course they might say that by Taylor's Theorem we can write, for small angles:
$$\ddot{\phi} + \frac{g}{l} \phi = 0 $$
However, in my ODE's course, I've been thaught that if we want to make a linear approximation of
$$\dot{\mathbf{X}} = \mathbf{F}( \mathbf{X},t)$$
arround an equilibrium point $P$, we have to use the Hartman - Grobman's Theorem (or a simplyfied version of it).
I agree that computing the jacobian matrix of $\mathbf{F}$ at $P$, $A$, will be the same as doing a first order Taylor in this case. But, there might be cases where the linear term is zero (the determinant of the $A$ is zero) and the theorem does not apply. If we think we are doing a Taylor approximation, we might be tempted to do a second (or higher) order approximation and, since it's not explicited in the theorem, might not always be true.
The question is: are we doing a Taylor approximation (it can be deduced from Taylor's Theorem / i.e. higher order approximations are allowed) or the correct way of approximating ODEs is the second one but we say we are doing "Taylor's approximations" because it's essentially the same thing for linear terms?