Is there a case when it is better to use the integral form of the Maxwell equations rather than the differential form?
The integral forms are useful in (typically static) situations where the charge/current distribution is symmetric enough that you can use a symmetry argument to replace the surface/line integrals with a simple product of a uniform field strength times an area/length of an imaginary closed "Gaussian surface" or "Amperian loop".
They are also useful for figuring out the far-field behavior of a localized charge/current source - e.g. we can use them to show that that, no matter how complicated a static localized charge distribution, very far away it produces an electric field of the form ${\bf E} = (Q/r^2)\, \hat{{\bf r}}$, where $Q$ is the distribution's total electric charge.
A friend of mine once attended a lecture titled "A Defense of the Integral Forms of Maxwell's Equations" in which a distinguished elderly physicist argued that the integral forms are underappreciated.
Take the interface conditions for electromagneticfields, e.g.
$$ \vec n\cdot \Delta\vec D = \vec\rho_s$$
i.e. the normal component of the $\vec D$ field is continuous if no surface charge $\vec\rho_s$ is present.
To derive that, imagine a box (or cylinder, prism etc) that goes through the interface parallel to the normal vector $\vec n$ (sorry, no pic yet). Then decompose the surface integral in Gauss' law such that
$$\begin{align} \newcommand{\oiint}{\oint\!\!\!\!\!\int} % sorry... \iiint \rho\,dV &= \oiint \vec D\,d\vec n \\ &= \iint_\text{top&bottom} \vec D\, d\vec n + \iint_\text{side}\vec D\, d\vec n \\ \iint\int\rho\,dh\,dA &= \iint_A\underbrace{(D_1-D_2)}_{=\Delta\vec D}\, dA + \iint_\text{side}\vec D\, d\vec n \end{align}$$
The last line uses that $\vec n_\text{bottom} = -\vec n_\text{top}$. Now let the height decrease to zero, thus $\iint_\text{side}\vec D\, d\vec n\to 0$ and define $\int\rho\,dh\to\rho_s$. The base area $A$ of top and bottom is arbitrary, thus the integrand is also equal and we just get the interface condition above.
The same principles apply as when you normally want the differential form of something vs. the integral form.
It's like asking when do you want to know the total distance and time of a road trip vs. what the speed was at a particular point.
With a uniformly charged sphere, for example, you can use the integral form to get the total flux at some radius, and then use that to infer what the potential is at any point because of the perfect symmetry of a sphere. Wouldn't work with a cube!
The analogy would be if you know you drove the exact same speed throughout an entire road trip, you could calculate what the speed was by dividing total distance / total time. Not so if there was a lot of starting and stopping!