If $X$ is a random variable, then the (cumulative) distribution function $F_X(x)$ of $X$ is ordinarily defined by
In the discrete case, if you know the probability mass function, you can compute the cumulative distribution function by adding. In the continuous case, if you know the density function, you can get the cumulative distribution function by integrating.
In more advanced work, generalizations of the cumulative distribution function are more natural than generalizations of probability mass function or density function. Indeed, in some situations, suitable generalization of these is impossible. But I will try to give a partial answer to your question in more or less familiar terms, without dragging in advanced matters.
Suppose, for example, that we are trying to make a probability model of the distribution of weights of people, or of the waiting time between consecutive buses. It is common and natural to use a continuous distribution as a model.
We are essentially never interested in the probability that the weight of a person is exactly $p$ pounds, where $p$ is a real number, like $50\pi$. For one thing, according to the model, this probability is $0$!
What we are typically interested in is the probability that the weight $W$ lies in a certain range, say greater than $200$ pounds, or between $140$ and $160$ pounds. That is information that can be easily obtained from the cumulative distribution function.
Even in the discrete case, we are often not all that interested, at the practical level, in the probability mass function. For instance, we are seldom interested in the probability that out of a population of $1000$, exactly $339$ are in favour of a certain candidate. We are more typically interested in the probability that a random variable lies in a certain range. The cumulative distribution function enables us to calculate this easily.
In later calculations, you will be interested, for example, in the distribution of a sum of two or more random variables. The cumulative distribution function will be very useful in this work.
Finally, let's make an analogy from basic physics. If we know the acceleration at all times, and the initial velocity, we can in principle compute the velocity at all times. So, one might ask, what's the point of having the concept of velocity? The analogy
is actually fairly close, since if we know the density function, then in principle we know the cumulative distribution function, and part of your question kind of asks what's the point of having the concept of cumulative distribution function.
One answer, for velocity and also for cumulative distribution function, goes as follows. These notions are important, both conceptually and at the practical level.