What does uncorrelatedness and independence imply?

Dear all,

I'm currently reading papers of statistical modelling. I encountered with the concepts of uncorrelatedness and independence. I understand the definitioins, but I am wondering what are the real effects they can make in statistical analysis?

For example, I have a dataset and I use certain technique (i.e. Principal Component Analysis) to separate the dataset into a set of uncorrelated vectors. On the other hand, I can separated the dataset into a set of independent vectors. What is the difference between these two sets of vectors? What property of uncorrelatedness and/or independence make such a difference?

Many thanks in advance. I do appreciate your kindly help.

Best wishes