Ax and axe are different spellings of the same word. Ax is the standard spelling in American English. Axe is standard outside the U.S. The distinction extends to compounds involving ax and axe. For instance, Americans use pickax and broadax, while English speakers elsewhere use pickaxe and broadaxe.
Examples
For example, these American publications use ax:
Legislature: Youth receiving centers face budget ax [Salt Lake Tribune]
Police say a call about Santa Claus running around a Buffalo Grove, Ill., neighborhood brandishing an ax turned out to be a student film project. [UPI.com]
And these British publications use axe:
Redditch United axe playing budget as manager leaves [BBC News]
Last month, the district council put forward proposals to axe travel tokens for its residents to save £88,000 a year … [Herald Series]
