Distributions and Random Variables I am confused about the relation between random variables and probability distributions in applications. 
Let $\Omega$ be a probability space. Often, $\Omega$ is some measure space that is turned into a  probability space by a probability distribution. Assume that $\Omega$ is $\mathbb R$ with a probability distribution $f$. Let $X$ be a random variable on $\Omega$. Then, the image of $X$ can be thought of as a probability space by defining the probability of a measurable subset $Y$ in its image to be
$$
\int_{X^{-1}(Y)} X \cdot f.
$$
Furthermore, now that the image of $X$ is considered a probability space, there exists a probability distribution $g$ that defines it, i.e., for each measurable subset $Y$ in the image of $X$,
$$
\int_{X^{-1}(Y)} X \cdot f = \int_{Y} g.
$$
What is an example (a real-world application) of when one cares about each of these objects, i.e., when would one want to defined a probability distribution on $\mathbb R$, then a random variable and then the probability distribution on the image of that random variable?
 A: I think the essence of your question is the following:

What is the point of using random variables if everything will boil
  down to a distribution in the end?

My personal feeling is that in the limited situation you are describing (when the measure space is $\Omega=\mathbb R$ and only allowed to consider one random variable at a time taking values in $\mathbb R$) there is indeed not that much to be gained by using random variables, other than convenience of notation.
However, working with random variables becomes essential on $\mathbb R^n$ for $n>1$, and I bet you are already encountering this case in practical applications without realizing it. For instance, any time you want to consider/compare two random variables on the same probability space, you are actually working with a random element of $\mathbb R^2$. This is the idea behind the concept of coupling. Indeed, just to describe the marginal law of a random variable $X$ given the joint distribution $(X,Y)$ requires random variables.
This is the point in the post where I will start using notations from measure theory, the rigorous foundation for modern probability - see here if you are not familiar with this notation.
Going back to the previous example, to describe the marginal law of a random variable $X$ given its joint distribution $(X,Y)$ with some other random variable $Y$, one must make use of a (measurable) mapping between measure spaces - in this case, $\mathbb R^2$ equipped with the Borel $\sigma$-algebra and a measure (perhaps given by a density similar to how you described, or perhaps not) and a mapping to $\mathbb R$ given by projecting onto the first coordinate.
Typically in a first-year graduate course in probability, one starts by learning measure theory then transitioning into probability theory - with the transition point usually being the moment that independence is introduced. Of course, independence can be studied purely in the context of measure theory under the name of "product measures", but much of the intuition is lost without thinking in terms of random variables. And this point is also the moment when the power/usefulness of working with random variables, and not just with distributions, comes into its own.
Now to wildly extrapolate the essence of your question to a much broader context. Many classes of mathematical objects can be categorized in a systematic way - not surprisingly, called category theory - with probability being no exception. Categories have objects, and maps between objects (one famous example being the category whose objects are sets, and whose mappings are functions). One can consider a category whose objects are probability spaces $(\Omega,\mathcal F,\mathbb P)$, and whose mappings are random variables (more precisely, random elements - since people typically reserve "random variable" for the case when the codomain is $\mathbb R$). From this perspective, the reason we care about random variables is the same reason we care about mappings in any category: even if the most "tangible" part of a category is its objects, all the most interesting stuff happens in the mappings between them. This is why (in my opinion) probabilists seem to talk a lot more about random variables than about probability distributions.
