Yes, you get different results if you don't normalize. My point was, I don't see why "you have to trivially normalize your data first" is a meaningful argument against anything.
Like all problems in statistics, it ought to depend on the specific task at hand. If there is some a priori reason to use the original scale (or a different re-weighting), it ought to be used. In general, PCA on correlation matrices is much preferred for exactly the reason you mention.