Learning a little about quotient spaces and I don't understand something.
(1) Compactify $\Bbb R^2$ to $[0,1]^2$ then glue sides to make torus. (linked post gives example of compactification)
(2) $\Bbb R^2 / \Bbb Z^2$ to make torus.
Are (1) and (2) the same or different? How so?
The two construction are different, though the resulting spaces are homeomorphic.
A compactification of $\mathbb R^2$ to $[0,1]^2$ is an embedding $\iota\colon \mathbb R^2\hookrightarrow [0,1]^2$ with dense image. That is, the image will be $\iota(\mathbb R^2)=(0,1)^2$. You then quotient $[0,1]^2$ by an equivalence relation given by $(0,a)\sim(1,a)$ and $(a,0)\sim(a,1)$ for all $a\in[0,1]$. In this construction, your original $\mathbb R^2$ gets embedded into the torus $[0,1]^2/{\sim}$ as the subspace $(0,1)^2/{\sim}$, where $\sim$ does nothing on $(0,1)^2$. This way you can think of the torus as $\mathbb R^2$ glued into the two loops formed by the boundary of $[0,1]^2$ after gluing.
In the case $\mathbb R^2/\mathbb Z^2$, which is a quotient by a group action, that is, the corresponding equivalence relation is $(x,y)\sim(x',y')$ if and only if $(x-x',y-y')\in\mathbb Z^2$, you don't embed $\mathbb R^2$ into the torus. Instead all the points which have the same fractional parts get mapped to the same point in the torus.