The difference of downsampling an image and smoothing an image?
Downsampling is the process of properly discarding every $(M-1)$ samples out of $M$ to reduce the signal sample rate by $M$. To avoid aliasing the signal being downsampled is initially lowpass filtered at a proper cutoff frequency before compression. The effect is a change of sample rate. Also the spectral positions of low frequencies are stretched to higher frequencies towards $\omega = \pi$. As a consequence, any spectra that passes beyond $\omega = \pi$ becomes aliased into mistreated bands, hence the requirement of a pre-processing lowpass anti-aliasing filter.

Smoothing is a more general thing. Whereas it may naturally refer to lowpass filtering in general, it may also mean more complicated things such as Kalman estimators which are also defined as smoothing operators. In the former case smoothing refers to getting rid of high frequency signal components, whereas in the latter case it refers to getting rid of randomness (usually of high frequency).