Hmmm. Tough choice.
In terms of aerial strength, SeeD does have the upper hand if we include Ragnarok, which was originally Esthar's. (There were 2 others, but they were used to move Adel's Tomb out into space and no one knows what happens to them. We can assume that they got infested by space monsters that killed off the crew inside.) The Ragnarok is armed with a 609mm charged particle cannon, two 152mm Multi-barreled Lasers and 2 retractable claws, but I doubt the claws can be of any use in actual aerial combat.
But the cannon and lasers have enough power to blow a hole into the Lunatic Pandora at point blank range, even after punching through its shield with relative ease. It is also capable of going into space and re-entering their planet's atmosphere.
Other than Ragnarok, we have the 3 mobile Gardens -- this is assuming that Galbadia, Trabia and Balamb are working together. (In an earlier match, Galbadia's Garden was included as a part of Galbadia, so maybe not..?)(Also, Trabia got nuked by missiles, but let's assume that they're not nuked.) As mobile shelters, the Gardens -- the buildings, do not have any shields or weapons, but is capable of ramming into things, as shown in the Battle of the Gardens.
ShinRa in terms of aerial fire power -- Has the Highwind. (We're assuming that the Highwind crew did not have a mutiny yet.) Not as fast as Ragnarok, but it is capable of chasing down the Ultimate WEAPON and ramming itself into it a few times without the ship suffering from damage. The FF Wikia states that it does not have any weaponry, but it is possible for it to have weapons fitted on (according to the weight) and as shown in Cid's Limit Break, it can shoot missiles. Can't go into space though.
Aside from the Highwind, we also have the Gelnika as a transport plane (unknown if it can use weapons) and military helicopters (can shoot missiles and has mounted guns).
As for trained human personnel...
ShinRa has SOLDIER (genetically enhanced by Mako and Jenova cells), Peacekeeping Troops (normal humans with military training) and the Turks (also human, but specially trained.) SOLDIERS have been seen making use of materia, but what about the Troopers and Turks? I want to assume that they can use materia though...
I won't go into too much detail about the strength of SOLDIER operatives since we already know them oh so well and their achievements, but it should be noted that the Turks -- if we're including the Turks from Before Crisis -- were able to take down Zirconiade, supposedly the ultimate summon and the Jade WEAPON, although it took them numerous battles across the planet to do it.
SeeD, although human, has trained combat specialists with the goal of taking down 'Mecia. According to the FF Wikia, Balamb Garden is the only one that trains SeeD cadets, conducts the SeeD exam and is the only one with SeeDs stationed at it? Galbadia's Garden focuses more on weapon technology. I might be wrong on this, but I think that Galbadia's Garden is the only Garden that doesn't have its students train with the Junction system/Para-Magic, which was why Irvine was able to remember Edea when others did not.
In addition to the normal SeeDs, there is also the White SeeDs -- an elite secret force of SeeDs. In the future, the dead bodies of SeeDs are wearing the White SeeD uniform, so I think they're the ones leading the fight against 'Mecia?
I kinda hesitate to use this term, but interestingly enough, both ShinRa and SeeD sorta makes use of child soldiers -- In SeeD, children as young as 5 can apply to be come a Cadet and between the ages of 15 and 19, can take the test to become a SeeD. With SOLDIER one can be as young as 14 years old provided they are strong enough to get in -- by extension I think that the same age limit can be applied to the Troopers, since Cloud failed to get into SOLDIER and joined the Peacekeepers instead. Not too sure about the Turks, but it was said that Cissnei was recruited to the Turks from an orphanage and trained from a very young age and is one of the youngest Turks around.
Some of the age brackets overlap -- SeeD has the age brackets ranging from 15 to 19 (I don't remember exactly if there was any SeeDs beyond the age of 19, but there might be some.) whereas SOLDIER has an estimated bracket from 14 to 25 (There might be older SOLDIERS, but I'm not sure. The oldest members that we know of is Angeal, Gen and Seph. Dunno about the Deepground peeps though.). For the PK Troops, we're not too sure about the age bracket but I'm assuming that they've been around before SOLDIER, so there's prolly some Troopers in their 30s and 40s, possibly 50s? Turks have no age brackets -- you leave in a body bag.
An average SeeD at most, had 10+ years of training (10, if they became SeeD at 15, 14, if they graduated at 19.) and a couple of years of field experience. SOLDIERs, excluding the Big 3, were normal humans without presumable training before they became SOLDIERs -- Cloud and Zack were normal countryside boys. Prior to joining SOLDIER, I assume that they train and after getting into SOLDIER, they are showered with Mako and Jenova Cells. (It's been a while since I last played, so do correct me if I'm wrong on this.) They train some more and do missions when not fighting in a war.
Other things/factors to take into consideration:
ShinRa
ShinRa's Machine Weapons -- The Black Widow, The Break/Barrier/Missiles/Any other variation that I'm missing/Machines, The Bee/Red Saucers, The Head series, The Crazy Series, the Sweepers and whatever the like I'm missing from the list. Yes, I'm aware that some of these machines go crazy in Crisis Core's missions but let us assume that in the fight against SeeD, these machines are working fine.
Something special to note: The Proud Clod. It was built to take down WEAPONS, but since we never see it happen, we have no idea if it is capable of taking down a WEAPON. hmm.
ShinRa's Science Mutations/Experiments -- mostly those crazy dogs with whips on their head. I forget if they're Hojo's or Hollander's or just some project they did for shit and giggles, but they can be weaponized. This is also assuming that Hollander did not defect from ShinRa.
http://finalfantasy.wikia.com/wiki/List_of_Crisis_Core_-Final_Fantasy_VII-_Enemies
The list I used to check for the machines and beasts.
Also, ShinRa kinda sorta has a Naval force? They do have ships and submarines...
The Sister Ray: Was used to shoot at the Diamond Weapon and take down Sephiroth's barrier at the Northern Crater. After it fired once, parts of it fall off the cannon and the blow back accidentally destroys a part of ShinRa HQ. I assume that because it was moved so hastily from Junon to Midgar, the Sister Ray was not properly attached to the building, and that the building is not structurally meant to hold up a giant cannon. Also, Seph, Gen and Angeal accidentally destroy the cannon, but only in a simulation, which may or may not be telling of how easy it is to destroy it.
As for SeeD...
The White SeeD -- They have their own ship, but it is not known if the boat has weapons.
Gardens also have assault boats, which they used during the Siege of Dollet.
There's also the Training Center which is full of monsters, but unlike ShinRa, it is unknown if they can weaponize them.
I'm pretty sure I'm missing some stuff about SeeD....
...This post is littered with assumptions since I haven't played VII/Crisis Core/VIII in a while, so if I got anything wrong, I'm sorry. OTL. All in all, I haven't really come up with a conclusion as to who would win in this post, but I might save it for a later post.
Edit: Derp. Forgot about Deepground and White SeeD.