The constructible numbers are those that can be achieved as lengths of line segments via compass and straightedge, starting with a segment of length $1$. The origami (constructible) numbers are those that can be achieved as lengths of line segments by folding paper, starting with a segment of length $1$.
We say a length has been achieved when it lies between two points occurring as intersections, of drawn lines and circles for the constructible numbers, and of intersecting folds for the origami numbers. I believe that we also allow points to be identified with origami by marking the image of an existing one when it's folded onto a new location (anyone more familiar with the axioms, please correct me if necessary).
It turns out $r\in\mathbb{R}^+$ is constructible if and only if it's contained in some chain of real quadratic field extensions of $\mathbb{Q}$ (Wentzel, 1837), and $r\in\mathbb{R}^+$ is origami-constructible it it's contained in some chain of real degree $2$ or $3$ extensions of $\mathbb{Q}$ (Haga, 1999). So the difference is that origami-constructible numbers are closed under taking cube roots.
I'm interested in seeing the simplest possible origami construction that realizes a cube root. By simplest, I mean minimal number of folds.
Copying from http://mathandmultimedia.com/2011/08/02/paper-folding-cube-root/ (but the diagram seems to be missing).
"Get a rectangular piece of paper and fold it in the middle, horizontally and vertically, and let the creases represent the coordinate axes. Let $M$ denote $(0,1)$ and let $R$ denote $(-r,0)$. Make a single fold that places $M$ on $y = -1$ and $R$ on $x=r$. The $x$-intercept of the fold is $\sqrt[3]{r}$."
You can't do better than one fold, nor better than degree 3, so this would seem to be the winner.
EDIT: Since the mathandmultimedia page leaves something to be desired, I looked for other duscussions of folding cube roots. Maybe https://www.researchgate.net/figure/Belochs-origami-construction-of-the-cube-root-of-two_fig2_233592288 will be helpful, as well as the link there to https://www.researchgate.net/publication/233592288_Solving_Cubics_With_Creases_The_Work_of_Beloch_and_Lill
There's a simple step-by-step $\root3\of2$ at http://www.cutoutfoldup.com/409-double-a-cube.php
An older question here on math.stackexchange is Solving Cubic Equations (With Origami)
The Hull article in the Monthly is also available at http://sigmaa.maa.org/mcst/documents/ORIGAMI.pdf
One more source for $\root3\of2$: http://www.math.ubc.ca/~cass/courses/m308-05b/projects/ayuen/doublecube.html