What is the difference between a'mixer' and a ' multiplier' used in modulation process?
For most purposes a mixer is a multiplier.

The issue regarding sum / difference frequencies is that when you multiply two signals of different frequencies, $\omega_1$ and $\omega_2$ say, then you get one component at the sum of the two frequencies and one component at the difference of the two frequencies:
$$
\cos(\omega_1 t) \cdot \cos(\omega_2 t) = \frac{1}{2} \left\{ \cos([\omega_1+\omega_2]t) + \cos([\omega_1 - \omega_2]t) \right \}$$



Response to comment

A frequency multiplier is a completely different kettle of fish. These are often highly nonlinear systems that take a signal $\cos(\omega_1 t)$ and produce $$\sum_{k=1}^{N} \cos(k \omega_1 t + \phi_k)$$

For example, a frequency tripler can be had by driving a transformer into saturation (so that it outputs something like a square wave) and low-pass filtering the result.