What is the point of countable vs. uncountable sets? I understand how to use these concepts and how to prove certain sets are countable or uncountable. However I don't get the point of it. What difference does it make whether a set is countable? People say that Cantor's proof that the real numbers are uncountable is a milestone of mathematics. Having read through the proof, I'm still struggling to understand what these ideas are important.
One important application is this: Consider the set of functions that take an integer argument and return an integer result.  It's not hard to show that this set is uncountable.  Now consider the set of computer programs.  This set is countable.  Therefore there are uncomputable functions—in fact most functions cannot be computed.

This is a pretty common type of argument in some contexts.
Sample a real number from $\mathbb{R}[0,1]$ as an infinite series of fair coin tosses.  Then, each sample is not only a refinement, also the start of a new sample.  The result being a rational may seem much less likely in the ergodic, but, sampling a rational sees sampling infinitely many copies of the terminus.  Sampling an irrational (in this manner) sees infinitely many different, thus that the irrational is represented less, of the balance that the rationals and irrationals are each dense in the reals.  It takes infinitely many fair coin tosses to sample a real number, that samples infinitely many, the sample is always plural.

Consider the function $\frac{n}{d}$ for naturals n,d with $0 \le n \le d$.  As $d \to \infty$, $lim_{n\to d} f = 1$.  Here $f$ ranges between zero and one and always includes zero and one.  Also, the differences $f(n+1)-f(n)$ are constant (uniform).  Then it is a CDF, uniform, with domain $\mathbb{N}$.

As that's in the countable it's a way to build notions of uniform probability from the natural integers, satisying various intuitions, and reasonably for application.  The point of the uncountable here is to see there are various ways to see what would be facts about these systems.

Matters of the infinite are relevant to mathematics, in the pure for what it is and the applied for what it is and does.

In terms of the applied and for practical purposes for real analysis:  additivity of measure is countable.  And, much work is done in formalizing analysis in the constructible and countable.  

It's good to know because other people believe it, to understand.  But, except for its own sake, there aren't results solely due the uncountable.  Results not of the uncountable are available via asymptotics, methods of exhaustion, and concrete mathematics.  Here for example, point discontinuities can be discarded in solutions due their vanishing without appeal to their zero measure.  And, the computable can be described in the countable constructible, the concrete mathematics is quite heavy.  For example, given a wager whether a random integer is even or odd, the asymptotic density of even integers in integers is $\frac{1}{2}$, as an example of that the cardinalities of those two set being the same does not reflect their densities being different.

So, there is much intuition about the real numbers, not all satisifed with the uncountable.  However, it is quite standard and in the general curriculum not knowing the general features as instructed and thus shared would be ignorant, regardless their eventual disposition or lack thereof.  

The ideas are important as an example of a way to consider the mathematically infinite.  It is important that their relevance is qualified, in their consideration.