Don't laugh. XD
The similarities between the two characters are so distinct and obvious that comparison is inevitable for anyone who's read/watched both series. I seriously thought when I first noticed Black Butler(more popularly referred to as Kuroshitsuji) that it was literally an anime adaption of Artemis Fowl. Other people have read Artemis Fowl and thought "CIEL HAS A CLONE??!?"
So I thought I'd outline some of the differences and analyze who would win. XD
This is the first similarity, and one of the most major, because if they didn't look so darn alike I don't think the comparison would have been as striking. The above picture of Artemis is a fanart, because there aren't really any good pictures of Artemis, and the manga style makes it more obvious. They are both from the British Isles and have black hair and blue eyes. Both of them lose/change/switch an eye resulting in two different colors.
They both start their respective stories at the same age(12). They both have parents who aren't present in their lives. (Ciel's are dead, Artemis's are kidnapped or on trips all the time). They both own their own businesses. Ciel's Funtom Company makes toys and restaurants and candy, and Artemis's makes electronics and technological wonder things .
They are both in charge of their households and businesses and constantly face being looked down upon by adults(and quickly setting those adults straight XD)
Also, they come from very wealthy families and own big shmancy mansions.
They both have butlers that they rely on for literally EVERYTHING. Ciel more than Artemis though.(more on this later) Both Butler and Sebastian are extremely talented fighters and cooks, and very devoted to their individual masters(although for entirely different reasons). Artemis and Butler are much more like friends(he calls Artemis by his first name) and
Artemis Fowl is a genius who describes himself as having twice the intellect of Mozart(but only 3/4ths the intellect of Da Vinci) and the highest IQ in Europe. Ciel doesn't claim quite as much, but he is extremely clever, accomplished, and smart enough to solve complicated murder mysteries and balance the drama and intrigue of London's elite. They both deal with the mafia and criminal underworld on a regular basis.
They are both very talented chess players. (If they were playing against each other I would bet on Artemis, although Ciel has never lost a game of any sort.)
They both play musical instruments: Artemis plays the piano and cello and Ciel plays the violin.
They both hang out with demons, fight zombies and evildoers, and get near-dead and revived on a pretty regular basis.
Both Artemis and Ciel are arrogant, confident, independent, and lonely. They both appear to be selfish but can be shown to have more compassionate sides. They are both ruthless, enterprising, calculating, cold, and appear to be driven entirely by greed. Here's where the similarities stop, however.
Artemis's confidence stems out of his own high self-esteem, hope, and trust in his intellect. He believes in securing a bright, sunshiny future for himself. He has hope in himself and in his brain and talents to get himself at the top of any situation.
Ciel, however, is completely devoid of hope. His confidence comes from his deal with Sebastian and his belief that he is sunk so low things really cannot get any worse for him. He places a low value on his own soul and life as evidenced by the contract, using himself as bait and willingly letting the queen use him for her dirty work. He believes that he is truly worthless, desperate, dirty, and defiled, and just playing a game with his time until he can achieve revenge.
Ciel also is more mature because of all the tragedy and horror he had in his childhood. He is much more patient and willing to offer wise advice to people. He operates on a deeper, darker plane than Artemis(who turns practically 100% "good") by the end of his series. He's "fallen" lower.
Ciel relies on Sebastian much, much, much more than Artemis relies on Butler. For the sake of the storyline, Eoin Colfer places Artemis in a separate situation from Butler on a regular basis, and he does just fine. Ciel, on the other hand, relies on Sebastian to dress him, give him baths, tie his eye patch, put on his shoes, and comb his hair. Every freaking morning.
Nevertheless, Ciel is more physically fit and combat-ready than Artemis and is armed at all times(he sleeps with his gun under his pillow.) He has a royal title which binds him to be loyal to the name of the queen and his family.
Artemis is more focused on science and technology(obviously, because of the setting of the two stories). He has more friends. He is also less likely to be loyal and form alliances than Ciel is, and he would never work for the queen or form a contract with Sebastian.
Besides, Ciel never:
“sold” the pyramids to a businessman in Cairo.
once thought that the only thing he needed to be a cartoon villain was an eye patch.(CHECK)
cut off a finger from two different people. Only once accidentally.
has been decomposed into his subatomic particles.
has been attacked by murderer crickets.
messed with the time continuum because of a
once had a little misunderstanding with a gorilla.
once had a little misunderstanding with a giant squid.
once had a little misunderstanding with an entire civilization of magical creatures
.worried about his mother buying him jeans when being stranded in the middle of the sea.
WAS THE NUT.
wears Armani underpants. And owns at least one red pair.
screams things like “Sell the Phonetix shares!” when recovering consciousness.
was dumped during the ceremony of the Nobel Prizes.
has lost, gained, switched and shared more than one part of his body.
And Artemis never:
joined the circus
traveled the world with a demon butler
had an epic fight on the top of the eiffel tower
made fifteen-minute chess metaphors
pointed a gun at somebody and pulled the trigger
So, the real question is, who would win?
I think it's inevitable that they start out as rivals, at least for a short time. They're both very similar and have a knack for ticking other people off. I don't think they could tolerate each other. XD So they'd definitely fight.
Most Ciel vs. Artemis posts I've seen are really Sebastian vs. Butler because they would invariably end up doing most of the fighting. If it came to that, I would really have to regretfully give the fight to Sebastian, because....invincibility is pretty hard to beat.
They key words are "if it came to that". Artemis, after duly observing and researching Ciel/Sebastian in action, would not go up against them directly to avoid the outcome of Butler having to face Sebastian on his own. They would definitely get some supernatural allies, maybe Grell and the Shinigami Haken Kyoukai.
Who knows, with his technology, maybe Artemis and Butler could hold up pretty well on their own. They did fend off an entire fairy army armed with time stopping machines and weapons of mass destruction. No doubt they would find some ancient way to break Sebastian off from Ciel and the pwn them each individually.
What about just Artemis and Ciel? In a battle of brawn, Ciel would definitely win since Artemis doesn't really like violence. In a battle of wits, definitely Artemis.
In general? Artemis. I just keep thinking back to the couple of episodes when Ciel didn't have Sebastian and ended up getting like ripped off, hungry and alone, his mansion caught on fire, and he was shot five times and the stomach and lay on the floor dying before Sebastian came and saved him. Artemis, on the other hand, broke into Spiro's Needle, escaped a bunch of hungry trolls, foiled Opal Koboi a bunch of times, and fought off a demon tribe without Butler..
You know what would be equally terrifying and epic? If Sebastian were contracted to Artemis. Now there's a scary thought. Look out, galaxy.
I'll leave you with this funny comic from Saracrista.