How are Principal Component analyses and Admixture analyses from a genetic alignment different? How are Principal Component analyses and Admixture analyses from a genetic alignment different? 

My understanding is that a PCA will take raw genetic differences across the entire alignment and plot them using dimensionality reduction techniques (single value decomposition for instance) whereas the Admixture analysis will plot likelihoods of sharing certain amounts of ancestry, based on allele frequency data. What I don't understand is how the two are fundamentally different. That is, isn't every variable position found in a PCA going to represent an allele used in Admixture?
Just to add to the great summary by Devon Ryan: admixture analysis tools are much more flexible than PCA (which is just a fixed mathematical operation), so they can be designed to incorporate LD patterns, phase information, different population evolution models etc.  

Here's a paper from the creators of fineSTRUCTURE and Chromopainter admixture tools detailing the action of both - it seems they provide some comparisons with PCA as well:
http://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1002453