One of the keys to system analysis in the notion of invariance.
A system can be seen as a box with inputs and outputs. To be able to understand outputs from inputs, or to predict outputs from inputs, you need to understand the rationality under the system.
Some crucial questions can be:
- are there "invariant" signals that are not modified, or only moderately so, through the system?
- can these special signals (sometimes called "root signals") be used to understand the behavior of arbitrary signals?
Linear and time-invariant systems use quite basic assumptions. Time-invariance is important in many cases: you don't want a sensor whose outputs, in the same conditions, change depending on the time of the day. Linearity or additivity is not respected everywhere, but many equations in physics are linear, or can be approximated, locally, by linear ones.
As said by @Matt L., only these two properties unfold a both "theoretically sound" and "practically efficient" framework. Root signals appear to be complex exponentials, they form bases of the space of signals. They can be used to interpret periodicity, the convolution inherent to the system can be turned into a simpler product in the dual frequency domain. Weighted averages become a central tool for defining filters.
And many more: you get a framework to discretized continuous signals and still deal with them easily (discrete Fourier transform), even faster implementations (FFT) and a wealth of related tools to deal with noise, nonstationarity, etc.
Disadvantages are LTI does not handle so well discontinuity and non-linearity (eg amplitude quantization, saturation). And since they are heavily related to least-squares and Gaussian noise, LTI is not the most robust to faults and outliers.
So far, I have no knowledge of many other systems so rich and so "simple" as the same time. The notion of root signals and structures has been well developed for median-like systems for instance, but might be not used a lot, except for median-filter design.
In image processing, linearity is often not verified. Mathematical morphology is another example of a powerful framework, not based on linear algebra, but on lattices. Apparently, they are somehow related through a Cramér transform. But this is a whole different story.