I would say the sheer manpower of the AOTP compared to most sized British and French armies. They also were trained to take 1.5x the number of steps most were trained while under fire, thanks to the Hardee Tactics manual that was a primer in American soldiery at the time.
The Army of the Potomac wasn't actually enormous compared to most British and French armies. The British never really deployed a field army in this period, as such, but they had enough support troops and disposable formations for five binary corps of binary divisions, which pretty much means a formation on par with the Army of Northern Virginia at Gettysburg. (20 inf brigades at 2,550 effectives, 5 cav brigades at 1,350 effectives, 40 batteries at ~250 effectives, total about 68,000)
The French army deployed in 1859 to Northern Italy was 128,000 men - larger than the AotP at Gettysburg.
As for taking steps under fire, there is a difference between what the manual says and the actual capability. One of the most glaring problems with American armies throughout the Civil War - Union and Confederate - is that while American troops can
stand under fire they have a great deal of difficulty
manoeuvering under fire. And if you're not moving at all then it doesn't really matter that you're trained to take more steps. (Look at all the times when an attack comes under fire and slows to a halt to fire back.)
The French are actually a case in point. French doctrine in 1863 was to
run into close quarters, and they did - straight under enemy fire at Solferino, which let them win the battle.
But the bigger question really is how much time the British are going to be spending under fire in the first place. British formations have artillery that shoots out to two miles with a reasonable degree of accuracy and every battalion has sharpshooters that can hit area targets (like enemy lines or columns) at half a mile with some shots - no matter what you count as "under fire", British troops can manage to put their enemy under it at much greater range.
These would help somewhat with the disadvantages, by making them faster than most European armies were under fire. But the British also still had implemented the Purchasing Program for advancement at this time, so the officer class wasn’t yet meritocracy. There was of course political appointments within the AOTP, but many of the officer Corps advanced due to merits on the battlefield (that and of course the need to replace lost commanders).
A lot of the officer corps got their start in the Army of the Potomac because they were well-off, or because they got elected to the position (i.e. a popularity contest). It's true that two years in there's been some meritocratic influence, but on the other hand the British have fought several wars over the last decade so the same applies.
All British officers have at least a minimum of professional training and thus competence, many of them have been promoted without purchase for action in the Crimea or the Mutiny or in China, and a substantial amount of the junior officers have actually had to pass exams to enter the army or assume their purchased rank. (That's before considering that the British have access to the graduands of Sandhurst and Woolwich, a pool 6-8 times the size of the West Point graduate pool which itself got split between the Union and the Confederacy.)
Compare this with the state in the Union army, where most officers have had to pick it up as they go along, and don't even have the benefit of being able to fall back on NCOs with 12+ years experience...
The British Army was also going through a transitional period as well. After the Crimean War, and especially after the Peel Coommission of 1858, led to a deal of reformation within the army. So the British Army in 1863 were not even half a decade into this remodeling.
Ah, this may be part of the issue. What you're probably not aware of is that the reforms after the Crimean War were done quite promptly; it's after 1863 that the army starts being
shrunk as part of a desire to reduce expenditure, and then Cardwell and Childers (again in the name of economy) actually make the army worse by reducing service time, reducing the number of battalions that can deploy overseas without problems, and otherwise cutting costs.
The British Army in 1863 is at the peak; it's large, extremely well trained and equipped, has recently been reformed in keeping with experience. The reform urge has not yet lost its way.
So I think a factor of the sheer size of the AOTP, compared to what England probably could have scrounged to face them (the 15,000 man force during the Trent Affair I feel is a good measure of what they could genuinely spare at this time)
Not really, unless your measure for how big the army the US could create is what they had one calendar month after Lincoln called for volunteers. The fact that there was a climbdown caused the cancellation of preparations.
In the 1861-5 period, there were about enough disposable British infantry battalions, home cavalry regiments and home foot, horse and garrison artillery battalions to raise five complete (British pattern) army corps. This is 60 battalions out of about 150 in the army in total, which means it's 40% of the army being deployed overseas; with about 230,000 men in the army in 1863 then that's about 92,000 men counting support troops etc. (so about 92,000 men AP.)
In the Crimean War the British deployed about 30,000 men in the initial movement to the Crimea out of a total army size of ~150,000; since then the British have expanded their militia system, created a volunteer system for home defence, and added another ~80,000 men to the total strength of the army. Their deployable manpower has increased considerably, because Crimea revealed a potential problem and the British responded to it.