I'd like to answer this question with one addition: Benefits outside the finance domain.
Financial mathematics, if you mean not only mathematical models but also computational finance in this encapsulation, is the ultimate numeric field to improve one's skills in statistics, math and computation (coding, algorithms etc.). The trick to finance is its rules are entirely social, not natural; a property many researchers choose to forget. Unlike natural sciences, you cannot fit a theory (theory in general sense) and expect to hold as a general rule. Though, many physicists thrive on quantitative finance jobs with their models designed to explain physics phenomena (e.g. Brownian motion).
In addition finance is oh so many layered. You can do analysis on micromarket structure (more playground for physicists), or you can simply do fundamental analysis of companies and invest in the long-term (oversimplified Warren Buffet).
In the past, economics and finance were more "social sciences" than a quantitative field. Bachelier's work in 1900 (brief, simplified note: He proposed stock returns are distributed in Brownian motion, later discovered and improved by Samuelson -in 1965- as log-returns). It is an important foundation of Black-Scholes model. His PhD supervisor joked to Markowitz that they might not give them the title because his work is no more economics (i.e. foundations of Markowitz Portfolio Theory). Nowadays, mathematical models are an integral part of modern economics and finance.
Nevertheless, they rarely hold for a long time. If you see Fama's work on Efficient Market Hypothesis (again early work is by Samuelson), he defends EMH with empirical examples (btw he slams bell-curve economists quite nicely -not as colorful as Taleb- and also has hints on HFT). In the subsequent but more boring and less known papers (EMH-II and EMH-III) he shifts his position (see Joint Hypothesis Problem). I have a technical report about that issue in option markets context.
One rule is for sure: market adapts. By extension, it loves to make surprises. Many models fail but we will keep on building new and more sophisticated ones. It will guide the decisions of investors, both informed and uninformed. I don't agree it improved risk management or correct pricing. There are two sides for each trade. Those who built models that predict mortgage market will fail against the risk models of the banks earned significant amount of money. With some stretch, I can agree it largely removed pure arbitrage, with the significant help of improved communication technologies. It gave us a methodological perception on the behavior of the market, just not always the true perception. Financial mathematics only help to improve the sophistication of the markets, this progress yield to many different methods and products (see the development of derivatives, HFT etc.).
Remember, trade can only occur if two parties disagree on the fair price of an asset. If we reach to an equilibrium state as many economists and finance scholars theorize, there will be no incentive to trade. Markets should change and adapt continuously in order to continue their existence.
Until this paragraph I tried to explain (with great confusion I worry) the underlying process of financial model progression and its effects on the general field. In my final say, I can recommend you the book An Engine, Not a Camera: How Financial Models Shape Markets Now the outside part.
Mathematical models built for physics play a large part in quantitative finance. Models used in finance can play a large part in other fields, such as education (which is also a social science that can benefit greatly from some quantitative approach). I can say statistical and mathematical models in education is largely outdated, unsophisticated and susceptible to large errors. It is possible to automate part of the learning process for the benefit of both educators and learners. You can see some examples in adaptive learning and notice how similar some algorithms are to the finance related models and equations.