Although I like the other answers to this question I think, there are some points which may be interesting to note and should get attention as well. Let me address each of your individual questions.
My question is, what are the benefits that both approaches contribute the most in finance?
First, I think , methodology in finance is not about being Bayesian or Frequentist but about searching for an answer using appropriate tools. As stated by @lehalle, its all about the data, the model and the objects you are after.
Financial datasets are, in general, not very large. You may have firm-month observations of thousands of firms, but as companies are usually not surviving for hundreds of years, time-series are relatively short. This leads to the problem that your parameters are not going to be estimated with large precision.
How can you handle this estimation uncertainty?
Well, either you go for frequentist standard errors, which may be unnecessary inflated as they are only driven by the number of observations, or you choose some well-motivated priors in order to include additional information which decreases your parameter uncertainty.
Trade-off here: You may impose priors which are not accepted by your peers or you may not be able to obtain some meaningful, significant results.
Often, the objects you are interested in are hard to capture by Frequentist methods. Stating empirical evidence without being able to asses the estimation uncertainty is worthless (in most cases). Let me give you one example where Bayesian Methods provide you with ways to computed Credible Regions where you would probably have a hard time to compute confidence intervals: Risk-management often cares about the Value-at-Risk, some quantile of your predicted distribution of future returns.
You may estimate the parameters of the return distribution which then yields you an estimated VaR. Can Frequentists tell you something about the uncertainty of this estimation? Well, the distribution of a quantile of return distribution is rather hard to capture. Fascinating world of Monte-Carlo Markov Chain (MCMC) allows you to simulate VaRs by sampling from the posterior distribution of the parameters and getting draws of associated VaRs conditional on the likelihood function. Bayesian Computation is a simulation tool which helps you to analyze otherwise extremely complicated statements. Trade-off here: Numerical approximations are not as handy as analytical close-form solutions, but at least there is some solution...You may also think about time-varying parameter models such as Stochastic Volatility Factor models, GMM and the like perform rather poor in this area.
- It really depends on your aim what fits best. Always keep in mind that results are interpreted differently depending on using Bayesian vs. Frequentist approaches. I personally think, Bayesian thinking is more natural in the sense that it overlaps with my subjective feeling for probabilities. Keep in mind that sometimes it is quite nice to have statements such as the probability of the CAPM being correct given the data is X percent (you may have a look in Doron Avramovs An exact Bayes test for asset pricing models).
What other areas in finance are Bayesian methods being used as industry standards?
This I don't know but you may find Rachevs book 'Bayesian Methods in Finance' useful.
Are there certain areas where one is favored than other? Should someone interested in Finance be gearing towards bayesian or frequentist?
I don't think anything should be preferred. Open mindedness for every method which helps to understand what is going on is the key. But: it is important to understand that there are differences in methodologies and some useless comments such as
Frequentist: "Bayesians can generate every result if they just torture the prior long enough"
Bayesian: "Asymptotic Theory is useless as data is always finite"