Differences between two normalization approaches I am currently try to normalize data.  But I am not sure the differences between $(x - \mu)/ \sigma$ and $x/\sigma$.  What are the advantages and differences of these two approaches?
 A: In the first one you center your data both by mean and variance; which means that your normalized data will be a 0-mean collection and have a variance of 1. In the second case, you simply normalize the variance so that you normalized collection will have a unit variance. I'll give you examples of those processes in the case of machine-learning algorithms.
Why is it important and when should I use it? 
Normalizing your data with variance (meaning creating a $x'$ that have a variance of 1) is really important most of the time because it scales every dimension of your feature vector to one so that their magnitude become comparable. It means that if you learn an algoithm that gives you weights on your data dimension (say $x=(x_1,x_2)$ and you learn a classifier with weights $w=(w_1,w_2)$) then you can compare those weights and tell that if $w_1 > w_2$ the impact of dimension 1 is higher than the one of dimension 2. I think it should be done most of the times!
Normalizing with mean is slightly subtler. It will change the interpretation of the intercept of a learned model, meaning if you want to predict $y=w_1x_1 + w_2x_2 + b$ then we've seen that if you haven't normalized your data with the variance then you wouldn't be able to compare the weights. If you haven't centered your data with mean you cannot give precise information over $b$ (the intercept). But if you have then you have: $b=\mathbb{E} y$ which is quite a nice result.
Caution with mean-centering: depending on the implementation or framework you use, centering with mean can be harmful because it can break sparsity and get your code to become very inefficient! So take care, sometimes it's better to prototype your algorithms with mean-centering for interpretability but not to do it with your whole dataset when deploying to production for performances. 
