What's the intuition for weak homotopy equivalence? 
Defintition: Two topological spaces $X$ and $Y$ are said to be homotopy equivalent
if there exist continuous maps $f: X \to Y$ and $g: Y \to X$, such
that the composition $f \circ g$ is homotopic to $\text{id}_Y$ and $g\circ f$ is homotopic to $\text{id}_X$.
Definition: A continuous map $f: X \to Y$ is called a weak homotopy equivalence if
it induces isomorphisms $\pi_n (X, x_0) \to \pi_n(Y,f(x_0))$ for all
$n \geq 0$ and all choices of basepoint $x_0$.

One can think of homotopy equivalent spaces as spaces, which can be deformed continuously into one another. However, I have absolutely no idea how to think about weak homotopy equivalences. How can one intuitively think about them?

Example: Warsaw circle; any map between the Warsaw circle and a point
induces a weak homotopy equivalence. However these two spaces are not
homotopy equivalent (link)

How can one intuitively see why we have weak homotopy equivalence, but not homotopy equivalence? Which spaces have (intuitively) the same weak homotopy type?
What are the properties that weak homotopy equivalences preserve? What are the properties that weak homotopy equivalence does not preserve, but homotopy ewuivalence preserves? Are there concrete examples of applications of weak homotopy equivalences?
Also, why bother defining weak homotopy equivalence? How do we benefit from it and where is it useful? Especially, what is the motivation for it outside of CW complexes (since for them, it is equivalent to homotopy equivalence by Whitehead's theorem).
 A: Let’s take it for granted that we are interested in spaces up to homotopy equivalence. But how can we prove that two given spaces $X,Y$ are actually homotopy equivalent? One option is of cause to provide an actual homotopy equivalence, but this might be actually quite hard to do (we have to specify a lot of maps).
There are a lot of functors associating to a space $X$ some algebraic object (usually a group, for example singular homology, or homotopy groups) in a homotopy invariant way in the sense that homotopy equivalent spaces have isomorphic associated groups. Since algebraic objects are easier to handle than topological ones (for example a map is injective iff the kernel is zero) we might ask ourselves, if we can reverse this process: Given a map $X\rightarrow Y$ of spaces, which induces an isomorphism of the associated algebraic objects, is it maybe already an homotopy equivalence?
The notion of a weak equivalence of spaces is precisely of this form. And indeed Whiteheads tells you that for reasonable good spaces (CW-complexes) this actually works. However there are obstructions for general spaces, as the warsaw circle and actual circle demonstrate.
A: You are asking a lot of questions in one. Please allow me to answer with a few "motherhood" statements that may help. What we now call topology (it was then called "Analysis Situs") arose during the late 19th and early 20th centuries when it became apparent that a very abstract take on geometry could have fruitful consequences. However, the nature of that abstraction was not God-given: general topological spaces (which can be axiomatised and understood in many different ways), metric spaces of various kinds, manifolds, algebraic varieties etc., etc. all seemed like useful objects to study and so they were and are. I believe there was some shared background idea that later became known as Weyl's program leading mathematicians to think about mappings between these objects that preserve certain properties.
So, all that said, my point is that the relationship between topology and its many branches remains the work of Man. Topology is full of definitions that exist to make the proofs go through; the best way to understand the reason for those definitions is to study the proofs. Probably the best motivation for the concept "weak homotopy equivalence" is to study the proofs that use the concept.
