The general representation of Maxwell's equations in vacuum is given by:
$$
\begin{align}
\nabla \cdot \mathbf{E} & = \frac{\rho}{\varepsilon_{o}} \tag{0} \\
\nabla \cdot \mathbf{B} & = 0 \tag{1} \\
\nabla \times \mathbf{E} & = - \frac{\partial \mathbf{B}}{ \partial t } \tag{2} \\
\nabla \times \mathbf{B} & = \mu_{o} \ \mathbf{j} + \frac{1}{c^{2}} \ \frac{\partial \mathbf{E}}{ \partial t } \tag{3}
\end{align}
$$
where $c^{-2} = \mu_{o} \ \varepsilon_{o}$ is the speed of light defined by the product of the permitivity and permeability of free space. We can take the curl of Equations 2 and 3 to find:
$$
\begin{align}
\nabla^{2} \mathbf{E} - \frac{1}{c^{2}} \ \frac{\partial^{2} \mathbf{E}}{ \partial t^{2} } & = \frac{1}{\varepsilon_{o}} \left( \nabla \rho + \frac{1}{c^{2}} \ \frac{\partial \mathbf{j}}{ \partial t } \right) \tag{4} \\
\nabla^{2} \mathbf{B} - \frac{1}{c^{2}} \ \frac{\partial^{2} \mathbf{B}}{ \partial t^{2} } & = - \mu_{o} \ \nabla \times \mathbf{j} \tag{5}
\end{align}
$$
where these are referred to as the wave equations for $\mathbf{E}$ and $\mathbf{B}$.
In cases where induced fields are much smaller than the applied fields, are there any terms in the Maxwell equations that can be neglected?
By "induced fields" I am assuming you are referring to electromagnetic induction, correct? If so, then you can take the terms $\frac{\partial}{ \partial t } \rightarrow 0$, which will reduce the problem to an electrostatic or magnetostatic one.
Note that you can still induce electrical charges under the electrostatic approximation, i.e., called electrostatic induction. It's really just a redistribution of charges. If you neglect this effect as well, then let $\rho \rightarrow 0$.