Homeomorphisms are the maps that preserve all topological properties: from a structural point of view, homeomorphic spaces might as well be identical, though they may have very different underlying sets, and if they’re metrizable, they may carry very different (but equivalent) metrics. Isometries are the analogue for metric spaces, topological spaces carrying a specific metric: they preserve all metric properties, and of course those include the topological properties. Thus, all isometries are homeomorphisms, but the converse is false.
Consider the metric spaces $\langle X,d_X\rangle$ and $\langle Y,d_Y\rangle$ defined as follows: $X=\Bbb N,Y=\Bbb Z$, $$d_X(m,n)=\begin{cases}0,&\text{if }m=n\\1,&\text{if }m\ne n\;,\end{cases}$$ for all $m,n\in X$, and $$d_Y(m,n)=\begin{cases}0,&\text{if }m=n\\1,&\text{if }m\ne n\end{cases}$$ for all $m,n\in Y$. It’s easy to check that $d_X$ and $d_Y$ are metrics on $X$ and $Y$, respectively.
Clearly these are not the same space: they have different underlying sets. However, if $f:X\to Y$ is any bijection1 whatsoever, then $f$ is an isometry between $X$ and $Y$. $\langle X,d_X\rangle$ and $\langle Y,d_Y\rangle$ are structurally identical as metric spaces: if $P$ is any property of metric spaces $-$ not just of metrizable spaces, but of metric spaces with a specific metric $-$ then either $X$ and $Y$ both have $P$, or neither of them has $P$. There is no structural property of metric spaces that distinguishes them.
What I just said about $X$ and $Y$ is true of isometric spaces in general: there is no structural property of metric spaces that distinguishes them. Considered as metric spaces, they are structurally identical, though they may have different underlying sets.
Isometric spaces may even have the same underlying set but different metrics. Consider the following two metrics on $\Bbb N=\{0,1,2,\dots\}$. For any $m,n\in\Bbb N$,
$$d_0(m,n)=\begin{cases}
0,&\text{if }m=n\\\\
\left|\frac1m-\frac1n\right|,&\text{if }0\ne m\ne n\ne 0\\\\
\frac1m,&\text{if }n=0<m\\\\
\frac1n,&\text{if }m=0<n\;,
\end{cases}$$
and
$$d_1(m,n)=\begin{cases}
0,&\text{if }m=n\\\\
\left|\frac1m-\frac1n\right|,&\text{if }m\ne n\text{ and }m,n>1\\\\
1-\frac1m,&\text{if }n=0\text{ and }m>1\\\\
1-\frac1n,&\text{if }m=0\text{ and }n>1\\\\
\frac1m,&\text{if }n=1\ne m\\\\
\frac1n,&\text{if }m=1\ne n\;.
\end{cases}$$
It’s a good exercise to show that $$f:\Bbb N\to\Bbb N:n\mapsto\begin{cases}n,&\text{if }n>1\\1,&\text{if }n=0\\0,&\text{if }n=1\end{cases}$$ is an isometry between $\langle\Bbb N,d_0\rangle$ and $\langle\Bbb N,d_1\rangle$. (HINT: Both spaces are isometric to the space $\{0\}\cup\left\{\frac1n:n\in\Bbb Z^+\right\}$ with the usual metric.) Yet these are clearly not the same space: metric $d_0$ makes $0$ a limit point of the other points, but metric $d_1$ makes $0$ an isolated point.
I don’t know of any general method for finding an isometry between isometric spaces; if you can recognize two spaces as being isometric, you probably already have a good idea of what an isometry between them must look like.
1 If you want a specific bijection, $$f(n)=\begin{cases}0,&\text{if }n=0\\\\\frac{n}2,&\text{if }n>0\text{ and }n\text{ is even}\\\\-\frac{n+1}2,&\text{if }n\text{ is odd}\end{cases}$$ does the job.
Best Answer
As Didier commented, if you draw $\Bbb Z$ on a blackboard, you will have some dots placed horizontally. As you move away from the blackboard, the dots will appear closer, and finally, when you are at a sufficiently large distance from the blackboard, you will effectively see a continuous line, i.e. $\Bbb R$. The space $\Bbb Z$ by itself is not very interesting, it is a discrete space. When we look at it from far away, it has some interesting properties. Note that $\Bbb N$ is not quasi-isometric to $\Bbb Z$ (Why?). The essential principle of large-scale geometry is that you look at some object, then go "far away" and have a look at it again. What interesting properties does this object have now? All metric spaces that look similar when you are "far away" from them can thought to be quasi-isometric. Quasi-isometric metric spaces have the same "geometric properties". Later on, you will study notions like hyperbolicity, which are preserved under quasi-isometries. These "geometric properties" are termed as quasi-isometric invariants. Quasi-isometries are weakened bilipschitz maps. They can be thought of as discontinuous maps over whose discontinuities you have some amount of "control".