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