Why is the Poisson Approximation to the Binomial Distribution Useful?
The short answer is that the Poisson approximation is faster and easier to compute and reason about, and among other things tells you approximately how big the exact answer is.
Here's a simple example: suppose you're trying to get something to happen in a video game that is rare; maybe it happens 1% of the time you do something, independently. You'd like to know how likely it is to happen at least once if you try, say, 100 times. Here we have $p = \frac{1}{100}, n = 100$ and so the binomial distribution gives us an exact answer, namely
$$1 - \left( 1 - \frac{1}{100} \right)^{100}.$$
But how big is this? Do you know off the top of your head? Is it, say, bigger or less than 50%?
The Poisson approximation answers this question quickly and easily: in this special case, it amounts to the approximation
$$\left( 1 - \frac{1}{100} \right)^{100} \approx e^{-1} \approx 0.368 \dots $$
which gives
$$1 - \left( 1 - \frac{1}{100} \right)^{100} \approx 1 - e^{-1} \approx 0.632 \dots $$
so we get that the odds are about 63% that we'll succeed at least once, which is bigger than 50% but maybe smaller than you might hope.
We learn something else too: the Poisson approximation tells us more generally that the odds of success are approximately a function of the product $np = \lambda$ only (which is the expected number of successes), so that e.g. if we had $p = \frac{1}{1000}$ and $n = 1000$ the answer would still be about 63%. This is valuable information and not entirely obvious from the exact answer, and knowing it saves you from having to recompute a bunch of binomials.
Sometimes $n$ can get large enough that it would actually be infeasible to calculate the exact binomial answer. For example, suppose $n = 10^{25}, p = 10^{-25}$; numbers this big regularly appear in physics or chemistry since Avogadro's number is so large. I can confidently say that the answer is still about 63% even though it is no longer feasible to exactly calculate $(1 - p)^n$ (just try it!). The funny thing here is that the larger $n$ gets the harder it becomes to exactly compute the binomials, but the more accurate the Poisson approximation gets; for numbers this large it is for all intents and purposes basically exact.
One of the common reasons to use a Poisson distribution is when you don't really have a good guess at what $n$ is exactly (though you're sure it's large) - to be precise, a Poisson distribution can be described as follows:
The Poisson distribution of rate $\lambda$ is the limit of the binomial distributions with $n$ trials and an expectation of $\lambda$ successes.
This is most relevant when you want to measure an idea of events occurring independently from some sort of continuous source of independent possibilities. Maybe you never quite get to the true limit in reality, but things like:
How many neutrinos do I detect in an hour in my giant pool of water?
is exactly the sort of scenario where there's not much better than a Poisson distribution - we could be talking about a billion billion possibilities to detect something, but with many fewer successes - and we don't know exactly how many to expect anyways. Similarly, other questions such as "how many rain drops fall on a patch of land" or even "how many customer respond to a billboard" all fall into this case where, well, the intuition is pretty much that there's so many chances for success that we may as well model it as a continuous process rather than as a discrete one for some large $n$.
Otherwise said: the Poisson distribution does have some abstract interpretation due to the limit and there are scenarios where that abstraction would seem to be an appropriate model. It's not a question of approximating something we know the exact answer for, so much as modeling something where there was no exact answer to start with.
If you want more pure math reasons, the Poisson distribution has nice properties - for instance, if you add two independent Poisson distributed random variables together, you get another Poisson variable - which is not true of binomial distributions - which is related to the fact that the generating function $\mathbb E[X^k]$ where $k$ is distributed by the Poisson distribution is just $e^{\lambda (X - 1)}$, which is about as nice as you can expected a generating function to be.
You can also check that the convergence of binomial distributions to the Poisson distribution is quite fast; the probability that a binomial distribution equals $k$ is $$\frac{n\cdot (n-1)\cdot (n-2)\cdots (n-k+1)}{k!}\cdot \lambda^k\cdot \left(1-\frac{\lambda}n\right)^{n-k}$$ where the Poisson distribution is $$\frac{\lambda^k}{k!}\cdot \left(1-\frac{\lambda}n\right)^n.$$ The difference of these things decreases in proportion to $1/n$ for any fixed $k$ - which means that this is actually a good approximation for very large $n$.