Ocher and ochre are different spellings of the same word, referring to (1) any of several earthy mineral oxides of iron occurring in brown, yellow, or red and used as pigments, and (2) a moderate orange yellow. The only difference is that ocher is the American spelling while ochre is preferred outside the U.S.
Both spellings are many centuries old, and historical Google Books searches (which can be inaccurate when it comes to texts before 1800) uncover more instances of ocher than ochre in texts from before the 19th century. But ochre was the primary spelling throughout the English-speaking world, including in the U.S., by 1850. Even today, ochre is fairly common in American English, appearing only a little less often than ocher.
Examples
Outside the U.S.
It was only a few seconds, that glimpse of ochre houses, sweeping stairs and La Barca fountain, but it was enough. [Daily Mail]
A stream of ochre water splashes on the windscreen, leaving streaks the colour of dried blood. [The Australian]
Here, too, was strong colour: vermilion, orange and fuschia teamed with ochre and beige. [Independent]
The ochre buried with the burned remains suggest that it may have been part of a ceremonial burial. [Toronto Sun]
U.S.
It is here, on top of a thin layer of flaky ocher-colored rock, known as schist, that the vine is at its most unexpected. [Wall Street Journal]
Old colonial buildings painted Venetian red, ochre, terra cotta, burnt umber, and vermillion compete with the brilliant sky. [Boston Globe]
We hit the color with the clay, a strong ocher color. [Washington Post]
Colors are popping – including orange, burgundy and ochre. [Houston Chronicle]