If both the Poisson and Binomial distribution are discrete, then why do we need two different distributions?
The Binomial and Poisson distributions are similar, but they are different. Also, the fact that they are both discrete does not mean that they are the same. The Geometric distribution and one form of the Uniform distribution are also discrete, but they are very different from both the Binomial and Poisson distributions.
The difference between the two is that while both measure the number of certain random events (or "successes") within a certain frame, the Binomial is based on discrete events, while the Poisson is based on continuous events. That is, with a binomial distribution you have a certain number, $n$, of "attempts," each of which has probability of success $p$. With a Poisson distribution, you essentially have infinite attempts, with infinitesimal chance of success. That is, given a Binomial distribution with some $n,p$, if you let $n\rightarrow\infty$ and $p\rightarrow0$ in such a way that $np\rightarrow\lambda$, then that distribution approaches a Poisson distribution with parameter $\lambda$.
Because of this limiting effect, Poisson distributions are used to model occurences of events that could happen a very large number of times, but happen rarely. That is, they are used in situations that would be more properly represented by a Binomial distribution with a very large $n$ and small $p$, especially when the exact values of $n$ and $p$ are unknown. (Historically, the number of wrongful criminal convictions in a country)
The binomial distribution counts discrete occurrences among discrete trials.
The poisson distribution counts discrete occurrences among a continuous domain.
Ideally speaking, the poisson should only be used when success could occur at any point in a domain. Such as, for example, cars on a road over a period of time, or random knots in a string over a length, etc. We are talking about infinitely many infinitesimally small trials, each having at most one success.
In practice, though, the poisson can be used to approximate the binomial under certain conditions, but it is only a rough approximation. Such as using the Normal curve in place of a Binomial under the right conditions.