The difference between the two is that the 'warped' space is what we observe (because observing is an electromagnetic process, subject to gravity), 'real' space is something we can only infer (or observe in photos like in the linked article)
We can know the Earth is round without any reference to Euclidean space or Euclidean coordinates- just trace out a big triangle and you could observe that the interior angles will sum to more than 180 degrees.
The real space is useless for most intents and purposes (since our interactions are bounded by the properties of light and gravity, and thus appearances is what guides us).
It seems important to make the conceptual distinction however, and not confuse reality with appearances, in this case as well as every other.