The solutions of linear equations can in general be described as a superposition of other solutions of the same equation. This makes linear equations particularly easy to solve and reason about.
Nonlinear systems are more complex, and much harder to understand because of their lack of simple superposed solutions. In nonlinear systems the solutions to the equations do not form a vector space and cannot be superposed (added together) to produce new solutions. This makes solving the equations much harder than in linear systems.
The necessary mathematical techniques only started to be developed in the 20th century.
Examples of nonlinear systems: