Why not always use a binomial exact test to compare two proportions instead of chi square? I am trying to figure out what test I should use in the following scenario: I know that there is a lot of room for improvement in a specific area at work - being extremely critical, let's say that sampling $52$ observations, $31$ could be improved. After instituting an improvement / QA program for six months, let me assume that out of a sample of $55$ cases, there are only $11$ with residual flaws. The two samples are independent. We are therefore comparing two proportions: $p_{\text{ initial}} =\frac{31}{52}$ and $p_{\text{ final}} = \frac{11}{55}$. 
Although the numbers are exaggerated, I still want to see if the two proportions are statistically significantly different, and I think I have a couple of options: I can run an exact binomial test to calculate the probability that the new proportion of flawed observations, $\frac{11}{55}$, would occur if the actual underlying probability remained $\frac{31}{52}$. Alternatively, I can run a chi-squared test.
The chi-squared is an approximation, and what I have read is that it is to be applied when the total number of observations is too high. This is clearly not the case in the example; however, playing with the numbers in R, I couldn't see any delay or problems with the results even after using numbers $>10,000$. And there was no indication of any normal approximation being used.
So, if this is all true, why shouldn't we always opt for an exact binomial test, rather than a chi square?
The code in R for the two test would be:
    # Exact Binomial Test:
binom.test(c(11, 55 - 11), p = 31/52, alternative ="less")

    #Chi-square Test:
prop.test(c(31, 11), c(52, 55), correct = FALSE, alternative = 'greater')

 A: I think what the OP is observing is that using the Clopper-Pearson method for calculating exact binomial probabilities can be done for very large  in this age of fast computing, whereas in the past when the sample size got large, it was easier to use a normal approximation which would be pretty accurate with or without a continuity correction.  A chi-squared approximation could be another way.  It is my experience that for very large sample sizes (1000 or more) the binomial test can be computed accurately and relatively fast.
The only drawback to using exact methods for discrete distributions is that for any given sample size $n$, there are certain values for significant levels that cannot be obtained exactly.  So if you try to do sample size calculations searching to achieve a certain level of power and you search for the minimum sample size to achieve that power, you might be surprised to see that going from $n$ to $n+1$ can result in a decrease in power. This problem I referred to as a saw-toothed power function.
You can see examples of this in my paper with Christine Liu titled: The Saw-Toothed behavior of Power versus Sample Size and Software solutions: Single Binomial Proportion using Exact Methods.  The American Statistician May 2002. You can find it quickly by googling the saw-toothed power function.  The same issue applies to confidence intervals.
An earlier paper by Agresti and Coulli was published in the American Statistician in 1998 which is a popular method for generating Binomial Confidence Intervals.  This and other methods for obtaining binomial Confidence intervals can be found in the Wikipedia article titled binomial proportion confidence intervals.
A: You state that you have read the chi-squared test should be used when "the total number of observations is too high".  I have never heard this.  I don't believe it is true, although it is hard to say, since "too high" is quite vague.  There is a standard recommendation not to use the chi-squared test when there are any cells with expected counts less than 5.  This traditional warning is now known to be too conservative.  Having an expected count less than 5 in a cell is not really a problem.  Nonetheless, maybe what you heard is somehow related to that warning.  
As @whuber notes, the two different tests you ask about make different assumptions about your data.  The exact test assumes that the probability (31/52) is known a-priori and without error.  The chi-squared test estimates the proportions for both before and after.  Notably, both of those proportions are treated as having uncertainty due to sampling error.  
Thus, the chi-squared test will have less power, but is probably more honest.  It may well be that the true proportion of flawed observations was considerably lower than 31/52, but it looked that bad by chance alone.  You certainly may test if the after proportion is less than 31/52, just as you may test the after proportion against any value.  But a significant result would not necessarily imply that the process improved following the QA program; you should only conclude that the proportion is less than an arbitrary number.  
