Short answer:
It offers some degree -- and in many cases, a greater degree -- of comparability between two types of data (different assets, returns, etc.)
Long answer:
You may already know this, but keep in mind that "normalization" can mean different things (see this question). There are various methods and purposes for normalizing data (financial or otherwise) but keep things in perspective. Normalize when doing so would be helpful for what you're trying to accomplish, and use a normalization technique that is appropriate. Linear regression has limitations, taking the logarithm has limitations, and so on. It's great to have a big toolbox of different data transformations, but part of that is knowing what to use.
As an aside, you're right that empirically markets have not exhibited normal returns. In fact, Mandelbrot explains in this article that the Pareto distribution is more realistic. It was published in 1963, but more recently he talks in this book about how the data have continued to demonstrate this pattern. The point is that you may read or hear about normalization techniques that rest on assumptions like normality that may not always be suited to the problem at hand. At the risk of editorializing, the assumption of normality has been subtly embedded in a lot of financial research and it may sometimes be misleading, so make sure to check the assumptions underlying what you're being told.