In 1912, Piet Mondrian did two memorable things: he painted a series of trees, and he dropped an "a" from his surname, which had prevously been Mondriaan. In later life, trees would become anathema to him. Spotting one through the Madison Avenue window of a New York collector in 1941, the newly arrived Dutchman shuddered and said: "You did not tell me you lived in a rural area." In fact, trees were already disappearing from his work in 1912, their branches bent into the black lines and angles that would shortly solidify as grids. For this, and for the new spelling of his surname, we have to thank Mondrian's discovery of Picasso, the great god of modernism.  Arriving in Paris the year before, Mondrian had been astonished by Picasso's work, then passing through that phase of cubism known as analytic. The Spaniard, installed in a dingy studio in Boulevard de Clichy, was turning out works such as Bottle, Glass, Fork, the most completely abstract images he was ever to make. Had he continued on this road – fractured, flattened planes, shapes spelled as hieroglyphs, a pared-down palette of browns and greys – he might have beaten Mondrian to Mondrianism. But he didn't.  Instead, analytic cubism was followed by synthetic cubism (which integrated collaged paper into compositions), then by surrealism, neoclassicism and a spate of other isms and forms that continued, unabated, until Picasso's death six decades later. Each step engaged more literally with the world, resembled it more clearly. It was Mondrian who was left to take synthetic cubism to its Spockishly logical conclusion – an art reduced to grids of rectangles painted in black, white and the three primary colours. In 1919, he coined a name for this work: neo-plasticism, whose gospel he was to preach to ever-deafer ears until his death in New York in 1944. He wanted, he said, "nothing specific, nothing human" in his art. He got his wish.  The diverging trajectories of the two men are neatly suggested by a list of their Paris studios. In 1912, the year of Mondrian's trees, Picasso moved across the Seine from Clichy to Montparnasse and Boulevard Raspail; and then, a year later, around the corner to Rue Schoelcher. From there, in 1916, it was off to Rue Victor Hugo, then back across the river to Rue de la Boétie in 1918. Picasso took on his last Paris studio, in Rue des Grands Augustins, in 1937, having worked all the while in ateliers around France, in Normandy and Céret. Bar a break brought about by the first world war, Mondrian, in the same quarter of a century from 1911 to 1936, had had one studio, at 26 Rue du Départ. It is now buried under Montparnasse Tower.  You can weigh up the meaning of this difference in two exhibitions this spring and summer. In Madrid, Picasso en el Taller (Picasso in the Studio) follows its peripatetic subject from 1906 until his death, 60-odd years, a dozen ateliers and countless stylistic twists later. In Liverpool, Mondrian in his Studios opens at the Tate in June. That the ateliers of the second title are plural and those of the first singular seems mildly ironic (the Tate show takes in Mondrian's workshops in the Netherlands, New York and, fleetingly, London), although it turns out to make unexpected sense.  What becomes clear from the Spanish show is that Picasso did only ever have the one studio, although, it being in his head, this followed him around from place to place. Picasso en el Taller doesn't actually tell you this, its signs instead breathlessly pointing out that such-and-such an object in this or that painting – a side table, a fruit bowl – really was in the atelier where the picture was made. It is left to the viewer to work out that this means nothing at all.  The real story of Picasso in his atelier is told not by coincidental bowls or rocking chairs, but by pictures such as The Studio, which appears in this show. In 1955, when The Studio was painted, Picasso and Jacqueline Roque were living at La Californie, a florid 1920s villa outside Cannes. The work's title is misleading – I would imagine intentionally so. Insofar as the atelier at La Californie appears at all, it is as a frame, an ornate barrier between two states of being. The Studio is a double portrait of a world and a world beyond – not of a place but of space, or rather the constraints of space.  By 1955, Picasso had been fighting these constraints for half a century, wrestling with pictorial logic, breaking it down and twisting it so that all sides of everything could be seen equally and at once. Given this, the space most immediate to his work – the studios in which it was made – was bound to become a site of anxiety. When Picasso's ateliers do appear in his pictures, it is often, as here, in the form of doors or windows. The Studio is an image of yearning, for escape into a world that seems, here at least, to have been painted by Henri Matisse. Forty years before, Still Life with Balcony, Door and Key, made at Rue de la Boétie, spells out the same anxiety. Picasso sets out to unlock the space of his studio as a series of Euclid-busting planes, but it never quite seems clear – is that door open or shut? – whether he has managed to do so.  And Mondrian? If Picasso's life after 1912 was spent looking out, then Mondrian looked more and more inwards. Like any artist's studio, the one at Rue du Départ had a large window to allow for natural light. Mondrian's first action was to block this with gauze curtains or blinds, visible in photographs of the time. If Picasso waged war on Euclidian space on his canvases, then Mondrian, pragmatic and Dutch, fought his in the studio itself.  Photographs by André Kertész and others show the atelier at Rue du Départ as rectangular, although it wasn't. One of the reasons the perennially broke Dutchman could afford the space was that its odd shape, a lopsided pentagon, made it hard to let. Undaunted, Mondrian fashioned a rectangle in this rhomboid, implying its sides with imaginary lines running between the backs of easels and his few pieces of furniture. The resultant trompe l'oeil was successful enough to fool not just the camera but visitors such as the painter Ben Nicholson, who went away with tales of a four-sided studio that in truth had never existed.  This was more than mere game-playing on Mondrian's part. One of the tenets of neo-plasticism – perhaps its central belief – was that the day would come when the distinctions between art, architecture and design would disappear, leaving the world a creative whole. This process had to start somewhere, and the Rue du Départ studio was it. Just as the black gridlines of works such as Composition B (No II) with Red or Composition with Yellow, Blue and Red – both in Liverpool – either do not quite reach the edge of the unframed canvas or go over it, so those of the rectangles in Mondrian's studio ran on out into space, infusing the world beyond with neo-plastic energy.  The Dutchman, stiff as a board and invariably dressed in pinstripes and patent leather shoes, was bewitched by the jazz that seeped through the thin walls of his atelier from the dancing school next door. If music could be osmotic in this way, then why not other kinds of art as well? With its black floor and white walls hung with moveable panels of red, yellow and blue, the studio at Rue du Départ was not just a place for making Mondrians. It was a Mondrian – and a generator of Mondrians.  What makes this tale of two ateliers so curious is that it is so short – short enough to make an interesting exhibition on its own. In 1913, Picasso and Mondrian were, broadly speaking, the same artist. By 1919, they had nothing in common at all.  • Picasso en el Taller is at Fundación Mapfre, Madrid, until 11 May. Mondrian in his Studios is at Tate Liverpool, 6 June to 5 October.
Picasso hopped from place to place, while Mondrian made his life's work at one address. Charles Darwent on  what their studios reveal about their art