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.
That being said, just because such an assumption may fail to hold with generality (either theoretically or empirically) does not mean it isn't useful for effectively navigating some specific problem from a mathematical or computational perspective. For example, if you have a model that's super-expensive computationally, and you can speed it up a bunch by making some simplifying assumption without significantly altering the results, perhaps doing so is worthwhile. (As long as you rigorously check that the results do in fact remain within a reasonable degree of similarity.)
