The difference between convolution and cross-correlation from a signal-analysis point of view

I am trying to understand the difference between convolution to cross-correlation. I have read an understood This answer. I also understand the picture below.

But, in terms of signal processing, (a field which I know little about..), Given two signals (or maybe a signal and a filter?), When will we use convolution and when will we prefer to use cross correlation, I mean, When in real life analysing will we prefer convolution, and when, cross-correlation.

It seems like these two terms has a lot of use, so, what is that use?

*The cross-correlation here should read g*f instead of f*g

Answer

In signal processing, two problems are common:

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  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.

  
  


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  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.