The difference between convolution and cross-correlation from a signal-analysis point of view
In signal processing, two problems are common:


What is the output of this filter when its input is $x(t)$? The answer is given by $x(t)\ast h(t)$, where $h(t)$ is a signal called the "impulse response" of the filter, and $\ast$ is the convolution operation.
Given a noisy signal $y(t)$, is the signal $x(t)$ somehow present in $y(t)$? In other words, is $y(t)$ of the form $x(t)+n(t)$, where $n(t)$ is noise? The answer can be found by the correlation of $y(t)$ and $x(t)$. If the correlation is large for a given time delay $\tau$, then we may be confident in saying that the answer is yes.


Note that when the signals involved are symmetric, convolution and cross-correlation become the same operation; this case is also very common in some areas of DSP.