In 2001, Henri Cartier-Bresson reflected on the long moment in the early 1940s when he had briefly considered turning from photography to film-making. "If it had not been for the challenge of the work of Walker Evans," he wrote, "I don't think I would have remained a photographer."

It's this quote that provides the epigraph for Photographing America 1929-1947, a fascinating book that focuses on these two masters of 20th-century photography. Intriguingly, Evans's photographs span the years of the book's title, while Cartier-Bresson's were all taken between the spring of 1946 and the summer of 1947. It is tempting, if not altogether true, to say that Evans is essentially an American photographer (the American photographer?) while Cartier-Bresson is essentially a European one (the European one?) who, for a brief but illuminating period, trained his outsider's eye on America.

Cartier-Bresson arrived in America from the newly liberated France in May 1946 to prepare for his first big American exhibition at the Museum of Modern Art in New York. That show did not actually open until the following February because the curators, believing that Cartier-Bresson had been killed by the Nazis while attempting to escape from a prison camp, were actually planning a posthumous retrospective when he showed up on their doorstep.

In the interim, Cartier-Bresson took two working road trips to the American Deep South. On the first, undertaken just after he arrived in America, he was accompanied by Truman Capote for Fortune magazine, who would later memorably describe the French photographer "dancing along the pavement like an agitated dragonfly, three Leicas winging from straps around his neck, a fourth one hugged to his eye … clicking away with a joyous intensity, a religious absorption". Unconsciously or otherwise, Cartier-Bresson was following in the footsteps of Walker Evans, who had roamed the American south back in the summer of 1936, when he was on assignment for Fortune with the journalist James Agee. Their story of the plight of three tenant farming families in Alabama during the Great Depression was rejected by Fortune – they thought it too bleak – but, in 1941, in extended form, it became the acclaimed book, Let Us Now Praise Famous Men. Evans's photographs have since become iconic images of the time.

Cartier-Bresson travelled south again in April 1947, this time spending 70 exhausting days with a friend of Capote's, a young American poet called John Malcolm Brinnin. It was Brinnin who brought along a copy of Walker Evan's American Photographs. That book, published in 1938 to accompany the Moma show of the same name, had precipitated a revolution in American photography. A move away from straight photojournalism and the still-predominant notion of the beautiful image, it was the most influential photography book of the time and remains a seminal work.

Ironically enough, Evans had been influenced by a Cartier-Bresson show that he had seen in Julien Levy's small New York gallery in 1933. The photographs in the show were taken on a lightweight Leica and, as Evans later acknowledged, they revealed to him the liberating potential of a hand-held camera that shot at high speed. "The photograph is the blade which seizes the dazzling instant from eternity," Evans later wrote, anticipating Cartier-Bresson's famous definition of a great photograph as the freeze-framing of "the decisive moment".

Photographing America - the book is the English version of the French catalogue for a touring show that was exhibited at the Cartier Foundation in Paris in December of last year - shows how each influenced the other, and how their separate approaches, and their subject matter, overlapped most when the great European photographer turned his gaze on an America that the older American had already made his own. Saratoga Springs, New York (1931) – a rain-soaked, wintry, tree-lined urban landscape shot from above – looks like a Cartier-Bresson photograph of a European city but is, in fact, by Evans. Cartier-Bresson's portrait of a partially-sighted hobo, entitled San Antonio, Texas (1947), looks like it might belong in Evans's Let Us Now Praise Famous Men.

Elsewhere, however, their differences are striking. Cartier-Bresson liked movement and, although there are several landscapes in the book, he tended to train his camera on people. In Evans's photographs, the people tend to be stationary. Sometimes they're looking stoically at the camera, at other times they are caught unawares, often sleeping or lying exhausted on the streets. Though Evans would go on to define America through its vernacular topography – billboards, road signs, shop fronts, highways – the images here tend towards the kind of heightened socio-realism that characterised his pioneering reportage with Agee.

Two similar but revealingly different images sit side by side in this book: Cartier-Bresson's Natchez, Mississippi (1947) and Evans's Tupelo, Mississippi (1936). Both are photographs of wooden buildings in the American South. In Cartier-Bresson's photograph, the building is pristine, but another wooden structure has collapsed in front of it. A telegraph pole stands in the foreground, dividing the image in half, and your eye is drawn immediately to a little boy on a bicycle in the right-hand bottom corner.

Here is the decisive moment, all right, but it is surrounded by myriad signifiers, both formal and social. On the adjacent page, Evans's wooden building sits slightly askew between two other rickety shacks, a plume of desolate smoke visible behind them. The image is starker, its formal mastery altogether more understated. It signifies poverty, pure and simple, even though there are no people present. Nothing here distracts from the sense of quiet desolation; everything here adds to it.

Cartier-Bresson's American images are among the bleakest he took. Time and again, the people in his photographs seem lost or marooned on the city streets. In Manhattan (1947), a man lies unconscious, a broken bottle oozing alcohol like blood on to the pavement. In another, Jackson Mississippi (1947), a life-size dummy with a black face hangs from a tree beside a sign that reads "I may be hanging myself for paying such high prices for cars".

Arthur Miller later wrote of the photographs that Cartier-Bresson took in America: "Since his is fundamentally a tragic vision, he reacted most feelingly to what in America he saw as related to its decay, its pain." Evans would surely have approved. When The Decisive Moment was published in America in 1952, Evans reviewed it for the New York Times, describing Cartier-Bresson simply but tellingly as "a true man of the eye". He could also, of course, have been describing himself. Each of them, in their separate ways, reinvented photography, and although neither of them believed it to be as elevated as painting or writing, they helped make photography the art form it is today.

• This article was amended on 22 July 2009. The original said that the Paris exhibition of the works discussed here was due to be held later this year. This has been corrected, and the connection between the book and an earlier French exhibition catalogue clarified.