The Poisson distribution is also called the "law of rare events" -- it is the distribution that counts the number of occurrences of an event given that the probability of the event is very small.
That should sound a lot like the binomial distribution. In fact, the Poisson distribution can be derived as a limit of the binomial distribution.
So what is the point of it? First, the fact that the limit exists gives us a lot of analytically useful results. The Poisson distribution is easier to work with than the binomial distribution. It is easier to compute the pdf and especially the cdf. Its generating functions have nice properties. Etc.
Second, in applications, the Poisson distribution serves in ways that the binomial distribution just cannot handle. Consider the case of radioactive decay. You're measuring the rate, using a Geiger counter, on a sample of hundreds of trillions of atoms. The binomial distribution is arguably applicable in this case, but are we really sure that atoms are "discrete" in the same way the integers are? (That is an empirical question, up to science to figure out) I would argue that without a priori knowledge, using the Poisson approximation is not an approximation of the binomial distribution. It is an approximation of the "real" behavior of radioactive decay, whatever that is.