Difference between Real Analysis and Probability Theory?

There is a huge difference. The key additions are the concepts of independence (of sigma-fields), conditional independence (given a sigma-field), and conditional expectation/probability (given a sigma-field), which don't play a central (if any) role in Real Analysis. Probability and Statistics without the concept of conditional independence are hardly possible, and definitely boring. In my opinion, Kolmogorovov's "Grundbegriffe der Wahrscheinlichkeitsrechnung" major contribution is the introduction of the general definition of conditional expectation (which depends on the Radon-Nikodym machinery). The importance of this concept in the development of modern Probability and Mathematical Statistics is hard to overstate. Take a look at "Probability with Martingales" by David Williams, and "Theory of Statistics" by Mark J. Schervish.

Maybe we can put the question in the context of a "mathematical model". Kolmogorov (and others) came up with a model for probability theory that involves a measure in the sense of Lebesgue. Great. Now we can use the tools of measure theory to study probability theory. But certainly there is no reason to call them "the same".

Similarly, in physics, there have been given certain mathematical models for phenomena in the real world. But it is important not to confuse the model with the phenomenon modeled.

Here is a quote I like:

THESIS 22: Those who seek a phenomenon which exactly follows a mathematical model, seek in vain.

(F. Topsoe, Spontaneous Phenomena, Academic Press, 1990)