I believe that the following post by @technocrat in this topic is not right.
I believe in this case he might be confusing pbinom with rbinom, as pbinom indeed gives same result over any number of replications.
pbinom(., n, p) calculates the CDF of Bin(n, p) distribution, whereas rbinom(., n, p) generates (pseudo) random observations from the same distribution. So, to get reproducible results, set.seed can be used. But as CDF uniquely characterizes a distribution, so pbinom will always return the same result.