The sample space is useful for easy examples when you are learning probability theory. However, when you have very complex random experiments it becomes very hard to describe the sample space. For instance, there is something called a "branching" process. A branching process is when you have a single-parent, and that parent has a random number of children, those children can themselves have a random number of children, ect. This process might terminate, but it can also continue indefinitely. Here is are some pictures that illustrate one such possible path,
In the first and second image the branching process terminated, whereas in the third image the process continued forever. The sample space $\Omega$ will be the collection of all of these trees. This is an absolutely enormous sample space which is very hard to describe.
Rather in probability theory we tend to work with "random variables". A random variable takes each sample point in your sample space and assigns to it a quantity. For example, $X$ can represent the number of generations of each tree. Then we can ask for $P(X=\infty)$, the probability that the branching process continues indefinitely. By using random variables, we make thinking of random experiments a lot simpler.
However, it is important to understand, from a theoretical viewpoint, all random variables are defined on some sample space. Therefore, a mathematician will say $X$ is a random variable defined on the sample space $\Omega$ of all trees. However, we tend not to go back to the sample space. The foundations of probability theory require a sample space, however, when we do computations/calculations there are ways to avoid the sample space.