Do working statisticians care about the difference between frequentist and Bayesian inference? As an outsider, it appears that there are two competing views on how one should perform statistical inference. 
Are the two different methods both considered valid by working statisticians? 
Is choosing one considered more of a philosophical question? Or is the current situation considered problematic and attempts are being made to somehow unify the different approaches?
 A: While this is subjective, I would say:
It is called the Bayesian/frequentist "debate" for a reason.  There is a clear philosophical difference between the two approaches.  
But as with most things, it's a spectrum.  Some people are very much in one camp or the other and completely reject the alternative.  Most people probably fall somewhere in the middle.  I myself would use either method depending on the circumstances.
A: I would imagine that in applied fields the divide is not paid that much attention as researchers/practitioners tend to be pragmatic in applied works. You choose the tool that works given the context. 
However, the debate is alive and well among those who care about the philosophical issues underlying these two approaches. See for example the following blog posts of Andrew Gelman:


*

*Ya don't know Bayes, Jack

*Philosophy and the practice of Bayesian statistics
A: I don't think it matters very much, as long as the interpretation of the results is performed within the same framework as the analysis.  The main problem with frequentist statistics is that there is a natural tendency to treat the p-value of a frequentist significance test as if it was a Bayesian a-posteriori probability that the null hypothesis is true (and hence 1-p is the probability that the alternative hypothesis is true), or treating a frequentist confidence interval as a Bayesian credible interval (and hence assuming there is a 95% probability that the true value lies within a 95% confidence interval for the particular sample of data we have).  These sorts of interpretation are natural as it would be the direct answer to the question we would naturally want to ask.  It is a trade-off between whether the subjective element of the Bayesian approach (which is itself debatable, see e.g. Jaynes book) is sufficiently abhorrent that it is worth making do with an indirect answer to the key question (and vice versa).
As long as the form of the answer is acceptable, and we can agree on the assumptions made, then there is no reason to prefer one over the other - it is a matter of horses for courses.
I'm still a Bayesian though ;o)
A: Adding to what Shane says, I think the continuum comprises:


*

*Firm philosophical standing in the Bayes camp

*Both considered valid, with one approach more or less preferable for a given problem

*I'd use a Bayesian approach (at all or more often) but I don't have the time.

*Firm philosophical standing in the frequentist camp

*I do it like I learned in class. What's Bayes?


And yes, I know working statisticians and analysts at all of these points. Most of the time I'm living at #3, striving to spend more time at #2.
A: I think Bayesian statistics come into play in two different contexts. 
On the one hand, some researchers/statisticians are definitely convinced of the "Bayesian spirit" and, acknowledging the limit of the classical frequentist hypothesis framework, have decided to concentrate on Bayesian thinking. Studies in experimental psychology highlighting small effect sizes or borderline statistical significance are now increasingly relying on the Bayesian framework. In this respect, I like to cite some of the extensive work of Bruno Lecoutre (1-4) who contributed to developing the use of fiducial risk and Bayesian (M)ANOVA. I think the fact we can readily interpret a confidence interval in terms of probabilities applied on the parameter of interest (i.e. depending on the prior distribution) is a radical turn in statistical thinking. I can also imagine that everybody is actually aware of the ever growing work of Andrew Gelman in this domain, as pointed by @Skrikant, or of the incentive given by the International Society for Bayesian Analysis to use bayesian models. Frank Harrell also provides interesting outlines of Bayesian Methods for Clinicians, as applied to RCTs.
On the other hand, the Bayesian approach has proved successful in diagnostic medicine (5), and is often used as an ultimate alternative where traditional statistics would fail, if applicable at all. I am thinking of a psychometrical paper (6) where authors were interested in assessing the agreement between radiologists about the severity of hip fractures from a very limited data set (12 doctors x 15 radiography) and use an item response model for polytomous items.
Finally, a recent 45-pages paper published in Statistics in Medicine provides an interesting overview of the "penetrance" of bayesian modeling in biostatistics:

Ashby, D (2006). Bayesian statistics
  in medicine: a 25 year review.
  Statistics in Medicine, 25(21), 3589-631.

References


*

*Rouanet H., Lecoutre B. (1983). Specific inference in ANOVA: From significance tests to Bayesian procedures. British Journal of Mathematical and Statistical Psychology, 36, 252-268.

*Lecoutre B., Lecoutre M.-P., Poitevineau J. (2001). Uses, abuses and misuses of significance tests in the scientific community: Won't the Bayesian choice be unavoidable? International Statistical Review, 69, 399-418.

*Lecoutre B. (2006). Isn't everyone a Bayesian?. Indian Bayesian Society News Letter, III, 3-9.

*Lecoutre B. (2006). And if you were a Bayesian without knowing it? In A. Mohammad-Djafari (Ed.): 26th Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering. Melville : AIP Conference Proceedings Vol. 872, 15-22.

*Broemeling, L.D. (2007). Bayesian Biostatistics and Diagnostic Medicine. Chapman and Hall/CRC.

*Baldwin, P., Bernstein, J., and Wainer, H. (2009). Hip psychometrics. Statistics in Medicine, 28(17), 2277-92.

