This,

is an equation. Meaning it has to be solved.

This,

is a function. It represents the relationship between the x-coordinate and y-coordinate.

---

Diffrence of square means,

and they can always be factored.

---

Why use perfect squares?

You mean why bring everything to perfect square in a conic? Because after you used perfect squares the shape of the conic is easily recognizable. And also it can be graphed easily by simply doing a translation of the axes.