What is difference between detrend and normalization?
You detrend data in order to get rid of the linear trend in your data, which might cause spurious regression, misleading evidence that there is some relation between variables.

Normalization means adjusting values measured on different scales to a notionally common scale. 
Example you give: Standard score $\frac{X - \mu}{\sigma}$ is just one possible method of normalization . It allows you to see where value lies in comparison to mean. 

In my experience you would only detrend data in order to create time series model. On the other hand, normalization is frequently used to compare previously not comparable statistics or detect anomalies (in case of standard score). As a result, usage really depends on the use case and you rarely use both at the same time.

Wikipedia is good enough source to clarify this unambiguity.