Howard Jones's Blue and Gray Diplomacy is really excellent on these questions, but in short, the geopolitical advantages did not outweigh the liabilities politically, economically, or diplomatically. The Confederates and their local allies tried to secure recognition, but Union diplomats were far more active (if only because of numbers and freedom of transit) and had a lot more leverage in terms that really mattered to the national security of European states.

In effect, any nation recognizing the Confederacy as a separate nation would be making war against the Union. (Secretary of State) Seward made this quite clear, and was angry enough when the British accorded "Belligerents' Rights" to the Confederacy (as did others) in terms of prizes, freedom of passage (see the Trent affair), the interning of refugees, and the legality of their actions on the high seas in defiance of the US government. But they were careful to claim neutrality in doing so, and the US blockade of southern ports was very much in keeping with wartime belligerents, not simple rebellion.

But recognizing the Confederacy during the conflict would give the stamp of approval to the breakup of the United States. That was not acceptable to Lincoln's administration, or to many Europeans. During the American Revolution, the recognition of the US by France and the Netherlands had been part and parcel of their entry into the conflict--but in that case they were taking advantage of an existing encumbrance to drive a wedge into a rival empire. And indeed some particularly in France and the UK were openly desirous of seeing growing US power stunted by fracturing the nation permanently (based on Parliamentary debates and French statements from 1863--not sure if the rhetoric was quite the same before). But the slavery issue certainly galvanized popular support for the Union (and electoral support was important to the British government, certainly), but within governments a few much less ideologically altruistic things also mattered.

First of all, the British certainly recognized the capacity of the industrial North to put ironclad ships into the sea if they wanted. If there was any doubt, Hampton Roads and the launching of New Ironsides ended it. A war, even if Britain had the greater force, would do enormous harm to British shipping--and the French would be the beneficiaries. Canada was also considered forfeit in case of war, as per Lord Russell (Foreign Minister for Palmerston). So the British wouldn't recognize without France. But France had its own issues: they had an adventure going on in Mexico thanks to the distraction between the States, where they'd put Maximilian I on the throne and propped him up with French aid (although he was popular among the Mexican elite, not so much in the countryside). They also weren't eager to get into a military conflict across an ocean while Central Europe was in turmoil, as it was in the 1860s. Emperor Napoleon III was strongest while France was at peace, as a diplomatic and military force in being. Both had to recognize, or neither would, but each had its own reasons for not doing so.

Added to all of this, French and British capital had enormous investments in the north. The importance of those connections is often forgotten, but shouldn't be understated. In case of war, they'd be severed, to the severe detriment of any party losing out. I'm not sure about the French public, but by late 1862 the British public was not likely to take kindly to a war to support a slaveholding economy. Early support out of worry about the loss of production (especially cotton) vanished as new sources in Egypt and India became available, and although European states permitted Confederates to buy things privately (so long as they didn't find out, or come under Union pressure to seize them, as they did with several blockade runners under construction) the governments remained openly neutral but effectively pro-Union. As long as they left the status uncertain, they lost nothing, but could change their stance if matters changed and avoid handing a geopolitical victory to their European neighbors at the same time--especially in the rivalry between Britain and France. Other European states saw no benefit to that kind of exposure, either.