Leonardo or Michelangelo: who is the greatest?

People are pushing at my back and trying to shove me aside, so they can pose, smiling, in front of the most famous smile in the world. Every photographic device the 21st century can invent, from the slenderest mobile phone to the most phallic telephoto lens, is being raised above the crowd to point at the woman isolated in her glass box. Her twilit painted world is jarred by flashbulbs as if by lightning.

This is the Louvre, in March 2010, and there are no prizes for guessing what painting is causing the fuss on this ordinary day. It seems, every time you see the Mona Lisa, crazier. As I cling to the crash barrier to stare as hard as I can, I can't deny it's a bit bizarre to see a painting idolised like a star at a movie premiere. But in truth, this is wonderful.

Here is a painting that is five centuries old – a relic from history, some would say. And yet it gets more visitors, from more places, than any modern work of art. This isn't to denigrate today's art, only to marvel at the timeless and universal genius of Leonardo da Vinci. Inevitably, the very fame of the Mona Lisa incites disappointment – having a pop at it is a critical vice – but I happen to believe she's worth every bit of adulation. The crowd is right.

If you were to look for an analogy with the fame of the Mona Lisa, the pushing and noise, only one other work of art comes close. Just as people make a beeline through the Louvre to find Leonardo's masterpiece, so do they queue right along a street in Florence, on a hot summer afternoon, to get into the Accademia gallery. The graffiti on its walls – "Don't bother, it's just a big statue" – doesn't put us off.

Two artists and two Renaissance wonders: Michelangelo's David and Leonardo da Vinci's Mona Lisa. Centuries after they were created, these are surely the two most renowned artistic objects in the world. They seem almost mirror images of one another – or rather, positive and negative: the woman who sits smiling, the man who stands grimacing.

That is no coincidence, because they were created at the same time, in the same city, by artists who were direct rivals, watching each other as intently as Matisse would one day watch Picasso. David and the Mona Lisa are monuments to a competitive standoff as direct and public and frenzied as today's Turner prize. It was the original and ultimate art competition, the Genius prize.

Over time, it became Michelangelo's habit to leave marks of the chisel on his works (the only signature most of them bear), as if his living, straining actions were fossilised in the chipped, unpolished surface of the marble. Entire works look like this: unfinished conundrums. Others are divided in their nature, with beautiful lifelike limbs and anguished faces bursting from pillars of stone, raw as it came out of the mountain. But there are no marks on the perfect youth. No chisel wounds blemish the masterpiece that made Michelangelo's name.

Luna was the Roman name for the quarry of Carrara, whose marble is as white as the moon's shining disc. The block Michelangelo stood in front of in 1501 had come from the quarry years before, had been "badly begun" by a semi-competent sculptor in the busy workshops of the cathedral, and then left there unwanted for 40 years. The tools with which the 26-year-old proposed to hew this massive lump of stone into a human shape were hammers and chisels, rasps and files and scrapers, and a wooden bow like an archer's whose string you could pull back and forth to rotate a drill.

With this simple technology, he had to excavate slowly into the 13ft-long marble slab, negotiating the clumsy damage done by its previous assailants, hoping his labour would not be wasted and that he would find the perfect limbs, the breathing sternum, the keen gaze within. The work was dusty, sweaty, back-breaking and secret, done behind partitions in the cathedral workshop so no one could spy on his measurements with the dividers, or watch him drill heart-shaped pupils into the statue's stone eyes.

It is impossible to picture this labour as you approach David today in the Accademia gallery; inconceivable, really, how he got from toil to miracle. Other works by Michelangelo may call attention to the struggle of creation – you walk towards the tall hero down a long avenue of unfinished bodies, striving to be liberated from formless stone – but this hero of youth is as absolutely himself as are any of the people walking around his plinth.

Stand far back, and his outline is a sharp drawing, as if Michelangelo had confidently mapped the shape in the air with pen and ink. The face, turned almost 90 degrees to look to the left, with its triangle of a nose, mountain outcrop of an overhanging brow and florid hair flying out into space, forms a scintillating profile. The proportions of the body are, from this distance, mathematically graceful. The measurement from the hair on the head to the fusillade of hair above the penis appears identical to that from genitals to toes. You can almost feel the weight of the body gracefully shifting on to its right foot, as the figure easily inclines its left knee forward, rolling its ribcage on top of its stomach to move its centre of gravity.

As you approach, this harmonious silhouette stays in your mind, yet also dissolves into glances and momentary impressions. The ridges and tensions of the immense chest high above you – the statue is more than twice the height of a living person, still higher because of the tall plinth – drink in nuances of shadow so that, up close, David is richly shaded: the belly button a pool of darkness, the nipples and ribs collecting delicate grey-greens. At his side hangs his gargantuan right hand – out of proportion, you suddenly realise, not just in scale but in the mesmerising, exaggerated attention to detail the sculptor lavished on it: those veins throbbing in the marble, those knobbly knuckles and wrinkled skin on the vast thumb.

Once you recognise the strangeness of this hand, the beautiful body Michelangelo has carved becomes still more alive. This, you start to comprehend – although actually you sensed it from that very first view along the avenue – is not some chilly, perfect nude. It is mobile, active, keen-eyed. The hand is the most radical instance of a quality that all David's parts possess: they are separate and slightly at odds with each other, like characters in a play. The statue may be finished as a work of art, but what it portrays is unfinished: a body still growing and changing. David contradicts himself even in his grace, because to be alive is to be contradictory.

Where David displays every muscle, his rival is respectably swathed. Her only action is to smile – to use what the anatomist Leonardo described coolly as "the muscles called lips". She is both mortal and goddess, smiling archaic personage and merchant's wife. Her pose has an eternal inevitability, as if she contained within her a serpentine column, revolving heavenward in a perfectly calibrated spiral: this effect of torsion means she is in energetic motion even as she sits still in her chair. The relief of shadow on her strong features gives her feminine beauty a masculine counter-life. She is a hall of mirrors, a shrine of paradox. Those who see the Mona Lisa's reputation as exaggerated are refusing to see how formidable her mixture of classical perfection and dreamlike ambiguity actually is; how much is in that smile.

The Mona Lisa dwells in a painted atmosphere so thick she might be suspended in tinted liquid. Reality melts in her world. Mountains dissolve, roads wind to nowhere. The power of this painting owes a lot to the strangeness and universality of its landscape, which feels like some kind of conclusion about the nature of life on earth.

Her portrait is drawn with shadows. The darks that deepen her features are so bold, you can lift them off and reproduce them as a black template. These shadows have the effect of diminishing the distance between foreground and background; the colours of the landscape bring it forward as her shadows draw her back. This heightens the psychological and poetic sense that somehow she contains grottoes and rocky recesses within her.

The tenebrous voids that darken her beauty make us unconsciously recognise that we cannot interpret this as merely a portrait with a landscape in the background. The vista beyond her, with its coiling road, arched bridge, rocks, rivers, lakes, mountains and sea, is as much part of her as she is.

The Mona Lisa – "Mona" or "Monna" being short for "Madonna", the reverent way to title a married woman in 16th-century Florence – started life as a portrait commissioned by Francesco del Giocondo, a textile manufacturer and merchant who had business dealings with Leonardo's notary father. But the picture of Francesco's wife that Leonardo showed his fellow-citizens in 1503 must have looked very different from today's unfathomable mystery. She must have looked like a real woman.

Leonardo worked on the Mona Lisa for years – perhaps until close to the end of his life. He never let the painting go, never handed it over to Francesco and Lisa del Giocondo. The poplar-wood panel was with him when he died. Leonardo's long and loving work – that and the smoke of time – created the dream picture we see today; it is impossible to see this as a "portrait" in any normal sense. As her obscurities deepened and her landscape ramified, so Lady Lisa was transfigured into a being of myth and fable.

Yet Leonardo's rhapsody really did start out as a portrait of a Florentine woman, and what amazed the first people who saw the picture was its brilliant verisimilitude. This, surely, is where she mirrors the lifelikeness of David who, though an ideal character from the Bible, was so closely observed in his anatomy that he seems almost to move.

"In this head, anyone who wanted to see how art has the power to imitate nature could easily understand it; for here were counterfeited all the minutiae that it was possible with subtlety to paint . . . the eyes had the lustre and moisture always seen in life . . ."

So wrote the artist and critic Giorgio Vasari in 1550, going into raptures for the curve of the Mona Lisa's eyebrows, the graceful nose, the mouth that "seemed, in truth, to be not colours but flesh". I feel the same way, standing under what seems the animate stone form of David.

Art in the 21st century happens in the glare of publicity and fame. New art is a public event, a media circus. It was like that in Renaissance Italy, too. Leonardo's new portrait got people talking when it was still just a sketch. The installation of Michelangelo's David in front of the city's government palace in 1504 suddenly unveiled a new star, nearly a quarter of a century younger than Leonardo but in the same incredible category of human genius. Their new works were self-evidently similar not just in quality, but in appearance and theme. The human individual had never been portrayed so convincingly before. The Mona Lisa's first admirers said she was so lifelike, there seemed to be a pulse in her throat; thus with David's almost-beating ribs. The power of these objects, then and now, is to seem alive.

Both are classical; that is, harmonious and proportionate – the two most absolute Renaissance assimilations of the Greek style. Yet both have a quirky strangeness that takes them beyond that heritage, into the richness of the human. As to which is greater . . . can we give an answer? Contemporaries could not, so they decided to take it further.

The Florentine government set up a formal competition between the creators of these startling works. Both were challenged to translate their individual figures into multi-figured history paintings – to splurge their creativity on the grand scale of frescos. We might think it absurd to look at such supreme human treasures and ask which artist is best – but that was what the Florentine republic tried to establish, by getting them to work at public expense on competitive battle paintings, Leonardo's Battle of Anghiari and Michelangelo's Battle of Cascina, for the Great Council Hall of the civic palace. In inventing these vast compositions, both artists went beyond the public perfections of David and the Mona Lisa to create what are arguably their most personal, and tantalising, works. Known today only through traces and memories after the original designs were torn apart, the painted images covered over by later works, these lost battles still burn the imagination. They added another layer of originality to the wonders we still admire in the Louvre and the Accademia.

By 1506, after two years of the competition, the city of Florence was confident it knew who was the best artist, the ultimate genius. The prize was nothing so small as money. It was to decorate Rome itself, to define the look of the Vatican and shape the future of art. Meanwhile the loser left Italy, crossing the Alps to sulk in France, taking his most famous painting with him . . .

Today, she smiles enigmatically at her swarm of fans. It is a strange defeat.

The Lost Battles: Leonardo, Michelangelo and the Artistic Duel that Defined the Renaissance, by Jonathan Jones, is published by Simon & Schuster on Thursday, £25. To order a copy for £19.05 including free UK mainland p&p, go to guardian.co.uk/bookshop or call 0330 333 68467

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