J. Robert Oppenheimer was scientific director of the Manhattan Project, and director of the Los Alamos laboratory. He was directly responsible for not only building the atomic bomb, but also for the choice of how it was used, signing off on its use against civilian targets without warning, and playing a key role in the decision as to what cities to bomb.

Albert Einstein's role in the making of the atomic bomb was co-writing and signing a letter to President Roosevelt in 1939 urging that the US government should be looking into the problem of nuclear fission. He did not argue or urge that a bomb should be built — only that the possibility be studied. He did not "sign off" on the bomb — he signed off on the idea that nuclear fission should be looked at, and it was Roosevelt who actually implemented the idea. He certainly did not argue that a bomb should be used, much less used without warning against primarily civilians. The other person who co-wrote the letter, Leo Szilard, explicitly argued (and circulated a petition) that the US should not use the bomb in such a fashion. He wanted the US to build a bomb — to use as a deterrent against a German bomb. There is a difference between building a bomb as a deterrent and building a bomb as a first-strike weapon against civilians.

So even just based on the popular understanding of the two, Oppenheimer is the one who bears more responsibility both for the building of the bomb and for its use. Einstein's role, at best, was in getting the process started, but at the earliest and least-specific stage, and in a way that did not urge either creating, much less using, such a weapon.

Separately, on as somewhat deeper historical level, it is worth pointing out that Einstein's letter did not actually lead to the creation of the atomic bomb. It led to the establishment of the ineffectual Uranium Committee, which made very little progress, with very little enthusiasm, towards developing nuclear weapons. The US program only became a bomb-building one after the Uranium Committee was taken over by others and its mission radically changed. This occurred not because of Einstein, but because of the MAUD report. If Einstein had never written his letter, the path of history probably would not have changed very much at all — the same people who took over the Uranium Committee and turned it into something else (first the S-1 Committee, then the Army-operated Manhattan Engineer District), probably would have been able to achieve the same ends even if the Uranium Committee had not existed. The Manhattan Project — the all-out effort to build and use the atomic bomb — did not begin until 1942, long after Einstein had been excluded from the work.

Credit, blame, responsibility, stigma, etc., are all tricky historical categories, especially with something as multi-faceted and multi-authored in its origins as the development of the atomic bomb. But by pretty much any measure, Oppenheimer was more directly connected with its creation and use than Einstein. I have argued elsewhere that even if Einstein had never been born, one would probably expect the atomic bomb to have been developed during World War II, because its intellectual, institutional, and political origins are actually quite separate from Einstein, despite the outsized role he is usually given in the story (because he perfectly symbolizes the "knowledge is power" trope of it).