The previous answer is very good and I have no dispute with anything that it says. It does a great job of looking at political factors in the United States that led African American men to get the vote while no women had that right. However, I think that it is important to look at deeper societal and cultural reasons why it was deemed acceptable to exclude women from voting.
It seems to me that what really caused this exclusion was sexism. It was harder for white men to swallow the idea of women voting than it was to swallow the idea of black men voting. So, where did sexism come from?
It is, of course, hard to know. But I would guess that it came about because physical strength was more important in primitive societies. It was more important because it was needed both economically and in terms of defense. At a time when practically all work was physically strenuous and warfare was more or less hand-to-hand, male physical strength allowed them to dominate those areas and, thus, society.
Those days were gone by 1865, of course, but they were closer than they are to us now. This meant that people in 1865 were much less able to consider women to be equal. They had simply not “evolved” as far as we have because they had less time and because their society still required much more physical labor than ours does.
I would argue that these factors are as important as the political factors so excellently discussed in the previous answer.
This is a complex question. One point that should be made is that black males could already vote in some states before the Fifteenth Amendment was passed. So there was ample precedent for it. Another issue is that the woman suffrage movement was quite marginal, though gaining momentum, in the aftermath of the Civil War. Elizabeth Cady Stanton's demand for the franchise at Seneca Falls was still by far a minority position in the United States.
But perhaps the main concern among advocates for African-American rights was that adding women to the Fifteenth Amendment would have made it less politically palatable to many Americans. They argued that the time was most propitious for granting black men the vote, and that the time for women's political equality would come in the future. There were attempts by woman suffrage leaders like Stanton to add language to the Fifteenth Amendment prohibiting discrimination based on sex, but they were never taken seriously.
Another final reason might be that once enfranchised, black men were almost certain to become Republican voters. It was less certain that women would, and so the dominant party focused its energies on black men (of course, black women could not vote either.) In any case, many advocates for women, most famously Stanton, opposed the Fifteenth Amendment on the grounds that it did not enfranchise women. A lifelong abolitionist, she nevertheless made appeals grounded in race, and even argued that enfranchised black men would be a threat to white women.