Now, because the normal distribution is a continuous distribution, you will probably compute an answer to arbitrarily many decimal places. But this does not mean the result is more accurate. In fact, the answer is always less accurate, because the binomial distribution gives us the exact result.
The issues of computing the exact result numerically in the binomial distribution, due to the large factorials, can induce a source of numerical error if you are not careful, but if you imagine yourself with a magical computer with infinite precision and memory, the binomial distribution will always return the exact error for problems of this type.