How exactly did you do your trials?

For each set of trials, I picked a p value between 0 and 1. Then, for each trial, I "flipped a coin" 14 times and saw if it came up heads 10 times. If it did, I then tested if the next two flips were heads. The total I got at the end (roughly .486) is the number of times it gets two heads in a row after getting 10 out of 14 divided by the total number of times it gets 10 out of 14. I ran 20,000 sets of trials, and each set of trials consisted of 20,000 trials. The code for the program (it is written in C) is here:

http://pastebin.com/mc2F5tgz
Somethings to think about: Does the actual parameter value that you use in your simulation make a difference on which method is preferred? Can you think of a way to analyze the difference between the Bayesian estimate and the frequentist method without necessarily having to do simulations? Which prior did you use? Does this make a difference?

I really have no idea, I just wrote a program to simulate flipping a coin and then compared the results to the results within the blog article I linked in my first post. The blog article says the prior distribution is uniform.

As for your question of whether or not Bayesian methods are better - it highly depends on what you mean by "better". The questions I ask in the previous paragraph hopefully shed some light on that and make you think about it a little more.

You say you don't want to compare them on a philosophical basis and that's alright (hopefully I gave you some ideas on how to compare them on a more quantitative basis) but personally the philosophical basis is the reason I prefer Bayesian methods.

That's fine, but I don't have the knowledge to evaluate them on a philosophical basis, and in the interests of time I can only learn one. Since there are clear quantitative differences between the two, I want to choose the best, and it seems like the Bayesian method won out here. But that's only with the tiny bit of info I have to go on.