For example, for a linear time-invariant system, its output is the convolution of the input and the system's impulse response, which I don't know how to put into differential or difference equations.
A (discrete) LTI system is described by $y[n] = h[n] \star x[n]$. If $h[n]$ has finite support (FIR) then the output is a linear combination of the input, which can be written as a difference equation. But, even when $h[n]$ hasn't finite support, the transfer can often be expressed in the form $y[n] + b_1 y[n-1] + \cdots b_{M-1} y[n-M+1]=a_0 x[n] + x_1 x[n-1] + \cdots a_{N-1}x[n-N+1]$
which corresponds to a filter with "$N$ zeroes and $M$ poles" (using the Z-transform, $H(z)$ is a rational function), and that's the general expresion of a difference equation (see eg. here). Often, but not always: most LTI filters that appear in signal processing have $H(z)$ rational, but that is not necessary. So, the assertion is only partially correct. It's correct if we either restrict to filters with $H(z)$ rational, or if we allow infinite-order difference equations, or if we interpret it as an approximation (a LTI filter can be expressed with arbritrary precision by a rational $H(z)$, etc).
For continuous-time systems, it's analogous, using differential equations instead of difference equations.