Interpretability
Both the statistician and client benefit from a statistical model that is easier to interpret. Bayesian inference allows easier interpretability than frequentist inference.
Frequentist confidence intervals are more difficult to interpret correctly than Bayesian credible intervals (otherwise known as probability intervals). Most people intuitively misinterpret frequentist confidence intervals as though they were Bayesian credible intervals.
Frequentist model fit statistics are more difficult to interpret in complicated models, and are generally incomparable across multiple models with different methodologies. Bayesian model fit statistics, such as the DIC (Deviance Information Criterion), allow valid comparisons within complicated models and across multiple models with different methodologies. It is harder to interpret frequentist model fit when the statistics cannot be compared.
Frequentist inference considers the data to be random and the model parameters to be fixed. Bayesian inference considers the data to be fixed and the model parameters to be random. Frequentist inference considers the data to be random, in the sense that it relates to long-run frequencies. Suppose that you are on trial for a crime, and that the prosecutor calls a statistician who presents a statistical model regarding your potential guilt. If the model used frequentist inference, then the conclusions are based on a long-run frequency of the data that extends into the future. This obscures interpretability, and if it were me, I would want the model to be Bayesian and not infer about potential future data, which is inappropriate.
Frequentist inference has potential problems in a statistcal model with a sample size that is too large and too small, due respectively to issues in hypothesis testing and exact vs. approximate estimation. Bayesian inference is unbiased in relation to sample size. Since sample size can bias the interpretability of a frequentist statistcal model, Bayesian inference is preferable.
Bayesian inference produces posterior probability distributions, while frequentist inference merely produces point estimates. The additional information gleaned from Bayesian inference allows more graphic explorations of a Bayesian model. One example is that rather than only considering whether or not a pair of independent variables are correlated, Bayesian inference enables the correlation of posterior probability distributions of the pair of independent variables, which is what matters most. Another example is that Bayesian inference enables the statistician to replicate the data, given the model, while frequentist inference allows only one prediction or estimation per datum. For frequentist inference, an actual value may be compared with a predicted value, given the model due to point estimates. For Bayesian inference, an actual value may be compared with a distribution of predicted values due to posterior probability distributions. This Bayesian "posterior predictive check" enables a better understanding of how the model relates to the data, answers more potential questions, and therefore eases interpretation.
Although there are many other reasons not mentioned here, Bayesian inference is preferable to frequentist inference because it allows easier interpretability of a statistical model.