It is amazing how little info or photos of Maranta species there is on the web.
I have a wild species from the rainforest, but from the 9 species existing in that area, I didn't find one picture on the web, only herbarium specimen of a couple.

Here you go, Myriam.

Maranta

Zuzu!

Could someone please explain to me why the red-veined Maranta and the variety without the red veining are both lumped under the same name "Maranta leuconeura?" I just find it a bit confusing, since they look so different.

Because Maranta leuconeura is a wild species of plant that's simply been brought into cultivation and there tends to be a lot of variability between individual plants in wild populations. Merely looking different isn't what separates one species from another, but rather the line is drawn by reproductive incompatibility/isolation.

So there isn't even a varietal distinction between them?

Apparently not.

Hmm. Odd that. But thanks for the clarification.

I am genuinely perplexed as to why you think that's odd.

To classify a population of plants as a distinct species, subspecies, or variety, they have to be genetically isolated from other populations, either because they are incapable of interbreeding with them (species) or they are geographically isolated from populations they would otherwise be able to breed with (subspecies and varieties). At least that's the idealized version of how it works, reality is often quite a bit messier. But that's the general idea.

The fact that some individual plants of Maranta leuconeura look rather different than others is really of no consequence unless at least one of the above conditions is met.

I'm not a plant taxonomist, nor do I know anything about the genus Maranta. But, maybe it would help to think of it this way. Are blue eyed people a separate species or even subspecies from green eyed people? Obviously not, and the reason why is that they are not genetically isolated from each other (they can and do interbreed). Apparently, it's the same situation with the red veined Marantas.

I don't find it odd that they are the same species, but it just seems odd that there isn't at least a varietal name to differentiate them. I think at one time the red-veined one was referred to as Maranta leuconeura var. erythroneura (and still is on some sites), but I see that is shown as a synonym for Maranta leuconeura.