Can all systems be described by differential/difference equations?
No the converse is not true in general. 

Take for example the discrete-time ideal lowpass filter with impulse response 

$$ h[n] = \frac{ \sin( \omega_c n) }{ \pi n } ~~~,~~~-\infty &lt; n &lt;\infty$$

which describes an LTI system but it does not correspond to a difference equation of any kind. Indeed $h[n]$ is derived based on the inverse discrete-time Fourier transform of the filter frequency response;

$$ H(\omega) = \begin{cases} {1 ~~~,~~~ |\omega| &lt; \omega_c \\ 0 ~~~,~~~ \text{ otherwise } }\end{cases} $$ 

$$h[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} H(\omega) e^{j\omega n} d\omega $$