As far as Population, Industrial Might, technological innovation and Cohesiveness the Alliance wins hands-down.
Humans:
The Humans of Stormwind are an almalgamation of all the Human Kingdoms of Arathor. There are the few survivors of the original kingdom of Stormwind from the first War, a majority of Lordaeronians who moved South as part of the rebuilt Kingdom, the Arathi peoples who fled Stromgarde’s ruin, Gilneans, and the few decent people of Alterac who survived that Kingdom’s destruction.
The total population is unknown, but potentially near a million humans or more. It is difficult to estimate, though, since the city isn’t truly scaled, and we don’t see any of the outlying farms/villages you would see in the real world for a Medieval city of Stormwind’s size.
Dwarves:
The least impacted by all of the events so far, unfortunately, they do have a relatively small population compared to Humans. They’re probably the one population on Azeroth, though, that has had an increase in the past decade due to accepting into Ironforge the Wildhammer and Dark Iron clans.
Their Industry has been growing steadily and has been unharmed by enemy action since the second war and a primed to create an incredible industrial revolution in conjunction with Gnomish ingenuity.
Gnomes:
A devastated people, but slowly recovering from their near total societal collapse thanks to the aid of the Bronzebeards and Ironforge.
Night Elves:
Prior to the War of Thorns, they were considered the most populous, if scattered, peoples of the Alliance. Having existed as a disparate nation for 10,000 years and having peoples stretching from the north to the south of Kalimdor. They were experiencing a cultural revitalization thanks to the presence of outside races and their inclusion into the Alliance averting what seemed to be a total collapse and decay of Nationalism bred by immortality and boredom.
After the burning of Teldrassil the Night Elf population was devastated, the extent of which is unknown just how devastated, though reports are that there are a great number of survivors and casualties may not have been as extensive as previously thought with reports of refugees from Teldrassil filling Stormwind and Goldshire.
Draenei:
It is unknown how many Draenei there are, but considering they arrived on Azeroth in a single inter-stellar ship, there cannot be very many. However, the carrying capacity of the Exodar is unknown but can be considered massive. This is due to the simple fact that we know that 5 Similarly sized vessels carried the entire Draenei population to Draenor (The three ships captured by Kael’thas in Outland, the crashed Oshu’gun, and the Exodar), and carried enough of their people that they were able to spread out amongst a major city the approximate size of Stormwind (Shattrath) as well as several villages and even a major temple complex (Karabor) in Shadowmoon valley.
So their original populaton prior to being turned into pavement was probably equal to Stormwind’s current population, but has now been sliced to 1/5th of that.
All told, the Alliance can probably comfortably field a mixed army of around 75,000 Soldiers without straining their resources and populations over-much. In an all-out total war situation, you’re looking at an army of ~ 100,000 humans, probably as many as 10,000 Dwarves, several thousand Gnomes, likely another ~ 100,000 Night Elves, and 20,000 Draenei.
All told, an army around 240,000 strong. As Azeroth is a primarily Feudal, with some industrialization, but largely Feudal/Medieval society the Alliance would only be able to field those numbers during the main Campaigning season of the year. After-which close to half or more of that Army will have to go home to harvest crops, or Stormwind starves.
The Alliance can probably sustain a Standing year-round army of ~ 75,000 - 100,000 soldiers.
The Horde… not so much.
The Orcs.
We don’t know how many Orcs came through the Dark Portal in the first War. But we do know that the majority of them were Warriors and male. Some females came through (most notably Draka), but they were a distinct minority. Since the Orc Horde was described as an unending tide of green it can safely be assumed that they arrived in the Hundreds of Thousands.
However, following the second war, the majority of them were killed or captured. Only two clans escaped total confinement, the Frostwolves, and the Warsong. The Warsong were few enough in number to be able to evade patrols, and the Frostwolves remained hidden until encountering the Stormpikes.
The remainder of the Orcs were captured and placed in Internment Camps by the Alliance.
There are 3 known major Internment Camps: Hillsbrad, Durnholde, and Hammerfall. There were likely more, but these are the only ones mentioned canonically.
The total population numbers are unknown, but it can be safely assumed to be several hundred, if not a thousand or more per camp, since the cost of the camps were an especially large budget item for the Alliance after the second War and a point of major contention.
At the start of the Third War, though, it is unknown how many Orcs perished in the camps due to abuse or simple lethargy caused by the withdrawal of Fel-magics with the closing of the portal.
We do know, though, that the Orcs themselves were few enough in number that Thrall was able to rally the Orcs held in the camps, break them out, and with the entirety of the Warsong Clan, transport their entire population on a small number of stolen Alliance Warships.
This gives us a likely starting population of less than 20,000 Orcs remaining, with perhaps less than 20% of these being female. Orc Genetic make-up and population viability is unknown, but this would actually spell doom for a Human Population, as I do believe that Humans requires a minimum population of 50,000 to maintain sufficient genetic diversity to ensure survival of the species. (I really need to remember where I read that.)
Additionally, there was sufficient room left over on the Orc refugee ships to transport the entire population of the Darkspear Troll Clan, who therefore cannot number more than a few hundred, perhaps a couple thousand.
The next major Horde Race are the Tauren, who are of sufficiently small number to be able to live comfortably on a relatively small butte in Mulgore, and the chief clan, the Bloodhoof, are small enough to be threatened to extinction by the Centaur and for the combined Orc and Darkspear forces of so small a number to be considerable reinforcements and saviors of their people.
All told, these three races of the Horde likely number, combined, near the 50,000 mark.
You next have the Forsaken. Again, no hard numbers, but they are primarily the raised fallen of Lordaeron, but they are of such a small number that they were never considered a major threat by Arthas, and so are likely a very small percentage of the Scourge forces raised from Lordaeron.
Their numbers are small enough that a relatively small military state of a few hundred Scarlet Crusaders were enough to cause them considerable trouble for many years.
They likely number at most in the 10’s of thousands.
Blood Elves, used to have a City and surrounding area of similar size to Stormwind, and likely had a population as numerous. Numbering over a million souls until the Scourge came.
It is canonically stated that they lost 90% of their people to the Scourge, leaving them as the most numerous, but still devastated members of the Horde at ~ 100,000 remaining population numbers.
Taking typical Medieval population percentages for Soldiers VS. Civilians we come out to the Horde being able to field around 35,000 Solders straining all of their resources, the majorty being Forsaken, with the next largest number being Blood Elves, followed by Orcs and then small numbers of Tauren and Trolls. Perhaps even as many as 40,000 with the Highmountain and Mag’har included.
The Highmountain, and Mag’har likely do not add considerably to the Horde’s overall population numbers, as the Highmountain have a home even smaller than Thunder Bluff and Mulgore, and the Mag’har are refugees who came through in a single portal on their own. They may be elite warriors, and numbering in the thousands, but not of a significant number.
The Horde also has no major industrial base beyond what Gallywix’s Goblins have built for them. Unlike Ironforge they have no great manufactories, and what industrial systems they had beneath Orgrimmar were devastated during the Siege in MoP.
The Zandalari are probably the single greatest military force now in the Horde, as the occupy 1/3rd of a Continent the size of the Greater Stormwind Area (Elwynn, Redridge, Westfall). But, similarly, likely has a population of similar size to 1/3 of the Humans due to this. Nazmir is largely unpopulated jungle, or holds primarily Blood Trolls, Vol’dun is populated primarily be Sethrak and Vulpera.
They can probably field up to 30,000 Trolls for war, but most of those numbers would be dedicated to their Navy.
All told, if my estimates are accurate, the Alliance can field a maximum army size of ~240,000 Soldiers.
The Horde can field a maximum army size of ~70,000 Soldiers.
Lore-wise, and based off of incomplete sources and knowledge since Blizzard doesn’t like restricting themselves by having concrete World-Building and “There will always be enough Orcs… Even though we’ve canonically wiped them out three times.”
The problem for the Horde, from a Strategy Game Perspective. Is that the Alliance holds more territory, has a higher population, better industrial base and access to effectively infinite Wood and Metal for their industry. They also have a space ship.
The Horde can’t break Ironforge’s industrial supremacy. And the Night Elves give the Alliance access to Infinite lumber to build ships with Wisps. Combined now with the Tide-sages of Kul Tiras giving those ships magical strength and maneuverability… The Horde don’t have the numbers to stop the Alliance from keeping them decisively engaged on all fronts and gathering their forces for a devastating push wherever they feel like it.
The Vindicaar also gives the Alliance the ability to definitively halt any war in a matter of weeks by simply driving out to the local asteroid belt, picking up a sufficiently sized rock with a magical tractor beam, and dropping it on a Horde City of choice, then using magic to remove the dust from the atmosphere before any major ecological damage occurs.