I am learning these two notions and I was convinced that homotopy equivalence is a weaker condition(thus more spaces can be homotopy equivalent to one another) than homeomorphism in describing relationship between two topological spaces. Thus, homotopy equivalence is used to determine whether two spaces are NOT homeomorphism by finding two spaces are NOT homotopy equivalent.
Intuitively, on the other hand, when determining whether two spaces are homeomorphism or not geometrically, I noticed that people check to see if they can "transform" one space to the other. For example, the author of the book I'm reading claims that hemisphere(boundary not included) in $S^n$ is homeomorphic to $B_1(0)$ by projection. Can we make a similar claim for homotopy equivalence? Or, is this approach accurate for determining whether two spaces are homeomorphic or not?
To verify both homeomorphism and homotopy equivalence you have a similar process: you must verify the existence of a function that satisfies certain properties. One common way of doing this is to construct the appropriate function (i.e. to write down a formula for that function), and then to verify the appropriate properties. So yes, verifying homotopy equivalence is similar to verifying homeomorphism. However, as you can see when you examine the definitions, it is similar but more complicated.
So to prove $X$ and $Y$ are homeomorphic, you need to construct a bijective function $f : X \to Y$ and verify that $f$ is continuous and $f^{-1} : Y \to X$ is continuous.
But to prove $X$ and $Y$ are homotopy equivalent is more complicated. You need to construct continuous function $f : X \to Y$, and $g : Y \to X$, and $F : X \times [0,1] \to X$, and $G : Y \times [0,1] \to Y$, and you must verify that $F(x,0)=g \circ f(x)$, $F(x,1)=x$, $G(y,0) = f \circ g(y)$, and $F(y,1)=y$.
The functions $F$ and $G$ are "homotopies", and they make rigorous the intuition of "deforming".