You never technically need to work in a frequency domain. However, doing so can render certain types of differential/difference equations much easier to solve. Especially straight-forward to solve using these techniques are linear non-homogeneous initial value problems. The Laplace Transform can make mincemeat of many such problems, and ends up being much easier to use than many other techniques. In addition, the frequency information of, say, a particular RLC circuit can give you valuable information as to what the circuit does to a particular signal. For example, you may want to construct a low-pass filter, so you'd need to know that the poles of the transfer function of the circuit (in the frequency domain) are in a certain location. Thus, you're going to have to work in the frequency domain to solve a lot of design problems like that.
Experience will let you know what problems you can solve using transform techniques and what problems you can't. Generally, nonlinear problems do not succumb to transform techniques (some exceptions are certain nonlinear PDE's that you can solve using the Inverse Scattering Transform method, but that's not likely something you'll encounter unless you get into nonlinear fiber optics and solitons).