Originally, Wayfinder’s Guide to Eberron stated that both high elves and wood elves are found among both the Aereni and the Valenar, so these groups are not subraces. Wayfinder’s Guide to Eberron does offer Aereni the option of a skill or tool proficiency with doubled proficiency bonus, and Valenar the option of alternate weapon proficiencies, but those modifications can be applied to either high elves or wood elves with no difficulty.
The most recent, most official release, Eberron: Rising from the Last War, might arguably change this, however. Sort of. The chapter headings read “High Elves: The Aereni” and “Wood Elves: The Tairnadal,” implying that Aereni are all high elves and the Tairnadal are all wood elves. But nothing in the actual text under those headings describes this at all. Plus there is another heading, “Elves of the Five Nations,” which notes that there are both high elves and wood elves among the elves living in Khorvaire.
What all this reflects is that “high elf” and “wood elf” aren’t really Eberron concepts; those come from other settings like Forgotten Realms. You can use the stats for either, but in-character no one would refer to you in those ways. Really, drow are the only elf subrace that is actually distinct in Eberron; other elves all get lumped together, and then divided by culture rather than by stats.
In detail, non-drow elves fled the continent of Xen’drik sometime just prior to that continent’s destruction by the dragons. They landed on a large island they named Aerenal, or “Aeren’s Rest,” after the prophet, Aeren, who led them there. So all elves in Eberron are either drow, who stayed on Xen’drik, or not-drow, who left.
Wayfinder’s Guide to Eberron states definitively that the drow were bred by the giant slavemasters to hunt down and kill the other, rebellious elves. In previous editions, and to my preference, this was just one theory among many: how exactly it came to be that all the drow are in Xen’drik, and all the other elves are in Aerenal or Khorvaire, is something largely lost to time. Even if it did turn out to be a giant breeding program, only the oldest of Aerenal’s undying councilors would know it.
Conveniently, Eberron: Rising from the Last War makes no mention of that theory, so it would appear I was not alone in my preference.
Anyway, for the rest of the elves, Aerenal became home. This is all of the non-drow elves. Though you can subdivide these elves into three groups, those divisions came later, and those divisions are not nearly so stark as they are for elves vs. drow.
The three types of non-drow elves in Eberron are
The Aereni, who have stayed in Aerenal.
The Aereni worship the Undying Court, which is a group of deathless (undead-but-good-we-swear) elves, including the prophet Aeren, who first figured out how to become deathless. The Undying Court wields some high-grade communal magics, that tie all (non-drow) elves together. The dragons have a long-standing, unexplained, low-level hostility towards Aerenal, with periodic attacks that have to be fended off by the Undying Court. These attacks are always rather limited compared to what the dragons could do, so why they let the elves live is as confusing as why they have a problem with the elves in the first place.
House Phiarlan and its retainers, who left Aerenal after House Vol’s destruction.
(And also various other elves, either singly or in small groups, who have emigrated to Khorvaire for their own reasons over the last 3,600 years.)
Dragonmarks were a big, surprising development for the entire world of Eberron, and Phiarlan’s Mark of Shadow and Vol’s Mark of Death were two of the earliest (only the Mark of Hospitality is thought to have appeared earlier). House Vol managed to draw the ire of both dragons and elves with the birth of the half-dragon, half-elf Erandis d’Vol, and was utterly destroyed. House Phiarlan decided that if the elves of Aerenal would turn on one dragonmarked house, they might turn on another, and left.
The majority of Khorvaire’s elf population comes from this event, as it included not just the Phiarlan family itself but myriad other allied families and so on. These elves have been in Khorvaire for a long, long time, and are thoroughly integrated into the Galifaran, that is, human-majority, civilization there. And any other elven emigres in the intervening years would find elves welcomed into Galifaran culture, and so likely integrate as well. Though still tied to the Undying Court (at least from the Court’s perspective; some members of the Court have descendants among Khorvaire’s elves), most of them prefer the culture, including religion, of their adopted home, and so most worship the Sovereign Host (unless in Thrane, where the Church of the Silver Flame is more popular).
The Tairnadal, also known as the Valaes Tairn or (by non-elves) the Valenar.
Not all of the elves who wound up in Aerenal were content to rest with Aeren; some sought wealth and glory by sailing north, to Khorvaire, where they fought (primarily) goblins and orcs and lizardfolk, establishing a nation in the southeast that they called Valenar. They were eventually driven off Khorvaire by the Dhakaani Empire, over 10,000 years before the start of a typical campaign. The descendants of these same elves would return to Khorvaire during the Last War as mercenaries hired by Cyre. They would then betray Cyre, claim Valenar as their own, and that nation would be recognized by the Treaty of Thronehold.
Importantly, the Tairnadal are still largely based in Aerenal. Tairnadal warriors who travel to Valenar to hold the territory and win fortune and fame are known as the Valaes Tairn, and live there for years at a time, but they return to their homes in Aerenal to raise families and so on. They are in the process of moving more of their infrastructure off Aerenal and to Valenar, but that is very much still a work in progress; remember that Valenar is a very new nation, particularly by elven standards. In particular, all of the famed Valenar horses in Khorvaire are geldings; all breeding stock, as well as all foals, are kept in Aerenal to prevent Khorvairans (particularly House Vadalis) from stealing some and breeding their own.
Since they are born and raised in Aerenal, and have for millennia lived there, the Tairnadal are not that different from the Aereni. They have reverence for the Undying Court, they have ancestors among the Undying Court, and they can still beseech the Undying Court for aid or advice if they like—though they largely don’t. Unlike the other Aereni elves, the Tairnadal do not see joining the Undying Court as the greatest success an elf can have in life, and thus do not see the undying councilors as their most honored ancestors. Instead, they exalt glory and combat, and their hero ancestors who fought for Valenar in ages past are the ones they look to most. They believe that elven heroes are reincarnated over and over, and that each of them is the reincarnation of one ancestor—an ancestor who may have well been reincarnated many times before, and thus have multiple lifetimes’ worth of heroics to emulate. Those who succeed in most carefully emulating their particular ancestor are believed to be able to remember their past lives; these honored warriors are known as the revenant blades.