What’s the difference between analytical and numerical approaches to problems? I don't have much (good) math education beyond some basic university-level calculus. 

What do "analytical" and "numerical" mean? How are they different?
Analytical approach example:

Find the root of $f(x)=x-5$.

Analytical solution: $f(x)=x-5=0$, add $+5$ to both sides to get the answer $x=5$

Numerical solution:

let's guess $x=1$: $f(1)=1-5=-4$. A negative number. Let's guess $x=6$: $f(6)=6-5=1$. A positive number.

The answer must be between them. Let's try $x=\frac{6+1}{2}$: $f(\frac{7}{2})&lt;0$

So it must be between $\frac{7}{2}$ and $6$...etc.

This is called bisection method.

Numerical solutions are extremely abundant. The main reason is that sometimes we either don't have an analytical approach (try to solve $x^6-4x^5+\sin (x)-e^x+7-\frac{1}{x} =0$) or that the analytical solution is too slow and instead of computing for 15 hours and getting an exact solution, we rather compute for 15 seconds and get a good approximation.
The easiest way to understand analytical and numerical approaches is given below:
pi=22/7 is the approximate value which is numerical
1/2=0.5 is the exact value means analytic.