For ordinary or partial differential equations, there is a practical distinction between weakly nonlinear systems where the "standard" solution methods (finite element, finite difference, finite volume, etc) for the corresponding linear system still work well, and strongly nonlinear systems where they may not work at all.
An example of this type of weakly nonlinear system in would be the heat conduction equation in a situation where the material properties (specific heat and conductivity) are smoothly varying functions of temperature. A standard solution procedure (e.g. Crank-Nicholson time marching) will probably work well simply by using the "best estimates" of the material properties, based on the calculated temperatures, at each time step.
On the other hand this straightforward approach is unlikely to work for heat transfer system that involves phase changes and latent heat, for example modelling the shape of the interface between solid and liquid phases as the system is heated or cooled. The numerical method would need to include latent heat (for example by reformulating the problem in terms of enthalpy instead of temperature) and also the fact that if the discretization of the material is fixed in space, the boundary between the phases will not coincide exactly with the discretization points.
When solving any set of simultaneous equations (not necessarily derived from an ODE or PDE) by an iterative method, a similar division between weak and strong nonlinearity can often be made.