It was Pollock, apparently, whom the American people took to their hearts - queuing round the block to see his drips and daubs and splashes, while the Rothko show was a quieter affair, in every way. The myth of Pollock as the live-fast, die-young, all-American hero has proved an enduring one, and rather more appealing than that of Rothko: the melancholic rabbi.
Both exhibitions have now arrived in Europe (Pollock at the Tate and Rothko at the Musee d'Art Moderne in Paris), from which distance it is rather easier to take an impartial view of their different achievements. The Pollock exhibition is an undoubtedly majestic and exciting event, but Rothko, too, emerges as a major figure and, setting aside what we know of the men, their work looks evenly matched.
The Rothko show begins in 1935 with a group of works much influenced by his friend and neighbour, Milton Avery: low-key paintings of people in subways and on street corners; a gentle sort of urban realism painted in subtle tones of grey and blue and beige. The selection in Paris is truncated compared to the American showing, but the point is made that Rothko made a quiet, figurative start.
The most striking thing about Rothko's early work is how strongly he, like Pollock, came to be influenced by Picasso and Mir in the 1940s. The two Spaniards loom large over the decade - their work was widely exhibited in New York at the time and their presence deeply felt. Pollock felt Mir lightly and Picasso with conviction, with Rothko it was reverse.
Three "Untitled" paintings from 1940-1941 show Rothko's absorption of Picasso - the Picasso of Guernica, with heads and body parts stacked in blocks and given a vaguely mythological look - but it was Mir who provided the strongest inspiration. Rothko wasn't ashamed to borrow from him directly in pictures such as Hierarchical Birds and most tellingly in Tentacles of Memory: a series of spidery black lines linking scratchy patches of red and blue, set against a background of grey-brown horizontal bands.
Let's be clear about it, Rothko had long since ceased to be a student: he was 42 when he painted Tentacles of Memory - too old to be plagiarising so shamelessly, yet it was through these Miresque paintings that he found a way forward. Common to all are background bands of colour and tone - horizontal divisions which clearly anticipate the forms of his classic style, waiting for the elimination of the loosely figurative shapes and swirls.
The rooms in Paris are organised chronologically to show a definite progression through these early works to the moment in 1947 when he abandoned the figure completely with the first of the pictures known as "Multiforms" - luminous fields of colour broken by floating shapes and soft-edged blocks. They are painted like watercolours, wet on wet, with thin layers of colour bleeding at the edges. Gradually, the floating blocks became more ordered, stacked one on another in vertical piles and simplified until the pictures found the form that became Rothko's signature.
Ironically, the simpler the pictures became, the more complex their effect. They seem simultaneously empty and full: they draw the eye in and radiate light and colour out. These are the contradictions at the heart of Rothko's paintings which provide such enduring mystery.
On a basic level, the colours are breathtakingly beautiful and frequently surprising. Combinations which ought not work become a kind of visual perfection, yet, perversely, Rothko used to deny the importance of colour in his work. He preferred the illusion that the paintings were about something deeper and spoke weightily about the painter's search for clarity: "the elimination of all obstacles between the painter and the idea, and between the idea and the observer". This is all very well, but it ignores the unavoidable truth that for the spectator - if not for the maker - colour is the language that they speak and which determines their response.
Rothko knew this, of course, but he also knew that no two people will respond to a given colour or colour combination in quite the same way. Certain groupings can be relied upon - red tends to passion, blue and brown to melancholy - but the final mood belongs to the individual, not to the picture. That said, the artist had something very specific in mind for each of these pictures, defined by the arrangement of colour and shape, frequently with the weight of the picture bearing down from above.
I'm not sure if there are degrees of success among these images; the forms hardly change from one to the next, though some move me more than others. Many of the most striking are those that seem tied to the natural world, a world of horizons and sunsets: an electric line of orange hinging land and sky in Untitled 1953 or a Turneresque vortex of stored light in Untitled (Yellow, Pink, Yellow on Light Pink) 1955.
The comparison to Turner isn't as wild as it might seem. "Pictures of nothing and very like" was William Hazlitt's criticism of Turner's late depictions of light and mist and it's as good a description of Rothko as I can think of. At the opening of the New York Museum of Modern Arts Turner show in 1966 Rothko was heard to joke "that chap Turner learned a lot from me" and it was no accident that the Tate Gallery became the recipient of Rothko's Seagram murals in 1969, such was Rothko's pleasure in the idea of sharing a roof with Turner.
The murals were initially intended for, and commissioned by, the Four Seasons restaurant in New York's Seagram building, but, having eaten there one evening, Rothko changed his mind. Undeterred, he was increasingly drawn to working in series and to the idea that his paintings be grouped and seen together in a single room. The 18 works that he painted for a chapel in Houston between 1964 and 1967 (better a chapel than a restaurant) are probably his greatest single achievement.
The Houston pictures aren't represented in the Paris show, yet the point is made that Rothko's palette darkened as the 1950s progressed, as rust purples turn to black in the Seagram Murals and blues to black in Houston. The progression from the yellows and oranges of the early decade to the total darkness of black on black, as it is presented here, seems inevitable.
Great claims have been made for Rothko's last series of paintings, the black-on-grey canvases painted in the spring of 1969 - seen by many as the last word of a tragic nihilist and the works which conclude this exhibition. They were intended as finished works (he held a party for the paintings to gauge the opinion of his peers), but I'm not so sure. They have none of his trademark finesse or necessary depth: the brushstrokes seem clumsy and the spatial arrangement unresolved. They are dark, certainly, but they have none of the intensity of the inky-black paintings made five years earlier.
Rothko's career was peppered with bouts of depression and inactivity - he was never, by all accounts, a very cheery person - yet even with all this blackness and the knowledge of his suicide in 1970, the final atmosphere of the Paris show is far from depressing. The overriding sense is contemplative and slightly sad, but it is a delicious sort of sadness.
Much has been made over the years of the so-called "spiritual" quality of Rothko's pictures and of their kinship to "religious experience". I'm not too keen on this sort of talk as such phrases rarely actually mean anything, but a roomful of Rothkos has an undeniably meditative mood. His paintings have an enveloping quality, each one like a world in itself, and to be surrounded by them prompts an unspecific ache of longing for something not quite grasped. The work demands participation: even now a lot of people claim to miss the point, but, in a sense, what you get from Rothko's work depends entirely on what you bring to it.
Musee d'Art Moderne de la Ville de Paris, 11 avenue du President Wilson, Paris. Until 18 April