Why use the normal approximation to the binomial? In school, I was taught about the normal approximation to the binomial, and it was suggested that I could use it effectively under some conditions, because it can be 'easier to calculate'. 
I understand how this could be more convenient if I were using paper tables. Are there still advantages to using the normal approximation when all my computations are done using computers? Is it easier to do algebraic manipulations or calculus using the approximation? What are some examples of the advantages?
I don't know what the right benchmark test would be, but perhaps this gives an idea:
> benchmark(rbinom(1, 1, .5), replications=1000000)
               test replications elapsed relative user.self sys.self user.child
1 rbinom(1, 1, 0.5)      1000000   3.593        1     3.476    0.156          0
  sys.child
1         0
> benchmark(rnorm(1), replications=1000000)
      test replications elapsed relative user.self sys.self user.child
1 rnorm(1)      1000000   3.724        1     3.564      0.2          0
  sys.child
1         0

 A: I know of no reason to use the normal approximation to the binomial distribution in practice. There are a variety of exact algorithms that are more than good enough for general use, and these are what you get when you use the binomial RNGs from R, SciPy, etc. The only good reason I can think of to discuss the method in a statistics class is that you can use it to illustrate the central limit theorem.
A: The central limit theorem provides the reason why the normal can approximate the binomial in sufficiently large sample sizes.  Sufficiently large depends on the success parameter $p$.  When $p=0.5$, the binomial is symmetric and so the sample size does not need to be as much as if $p=0.95$, in which case the binomial could be highly skewed. Also, you get a better approximation when the continuity correction is applied.
Regarding your question about calculating binomial probabilities on the computer, the computer can calculate these probabilities quickly and therefore you really don't need a normal approximation.
