For an aircraft, what are the differences between nonlinear and linear equations of motion?
Equations of motion (EQM) define how a rigid-body accelerate and rotate given any arbitrary external forces and moments. They are no more and no less than Newton's Second Law. When expressed in body-frame, the EQMs are:

$$\bf{F}_B = m (\dot{\bf{V}}_b+\bf{\omega }_b \times \bf{V}_b)$$
$$\bf{M}_B=\bf{I}\dot{\bf{\omega }}_b$$

where $\bf{F}$ is body forces, $\bf{M}$ is body moments, $m$ is mass, $\bf{V}$ is inertial velocity, $\bf{\omega}$ is angular velocity, $\bf{I}$ is moments of inertia, and $\times$ is the cross-product operator.

The above are vector equations, which correspond to 6 scalar equations. At this point, there is no concept of aerodynamics. Notice how the equations are inherently nonlinear, so long as there are body forces/moments that depend on the state variables (e.g. velocity, orientation, etc). This is true for most forces, with the notable exception being gravity.

Aerodynamics is nonlinear in that aerodynamic forces and moments do not change linearly with flow incidence, rate of rotation and velocity, except if the changes are small. In general, quasi-steady aerodynamic force is a nonlinear function of flow incidence, rate of rotation, Mach number, Reynolds number, etc. To capture these, N-D lookup tables constructed from estimated/measured data are generally employed in simulation models.

Quasi-steady assumption itself may not be adequate in some flight conditions, such as during rapid maneuvering or stalled flight. In these circumstances, the description of aerodynamics may need to be time-variant.