Differences between factoring an equation and dividing one

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I've been browsing the Math exchange for some time (as well as other sites) looking for an answer in regards to:

"Are there any fundamental differences between factoring an equation, and dividing one?"

I found a decent answer here, but it fails to elaborate in depth on what these differing properties are.

For example, we can factor the equation

2x^2 + 4x - 8

By pulling out a 2, getting x^2 + 2x -4

However, is that fundamentally different in any way from

(2x^2 + 4x - 8)/2

Thank you for any help.