The binomial test is an exact test base on the binomial distribution while the proportion z-test uses the normal approximation.
For a small sample please use the binomial test.
|Binomial distribution - the probability for event within each group is identical|
|Expected difference d between the populations's average is known|
|p the sample probabilities or x the number of successes|
|n - Sample size|
The following R code should produce the same results:
The prop.test uses the Chi-squared statistic with one degree of freedom, which is exactly as using the Z test: z2=χ2(1)