Ax and axe are different spellings of the same word. Axe is standard in varieties of English from outside the U.S. Axe also appears in American English, but the newer spelling, ax, has gained ground over the last half century and is now more common.
The distinction extends to compounds involving ax and axe. For instance, Americans often use pickax and broadax, while English speakers elsewhere use pickaxe and broadaxe.
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]
But it’s just as easy to find examples of axe in American sources:
Mr. Hutchinson clutched the axe he always stores in his attic and cleared an escape hole for the four people and three pets. [Wall Street Journal]
But Ryan’s budget axe comes down hardest on Medicaid — not Medicare. [Washington Post Wonkbook blog]
Axe is used throughout the non-U.S. English-speaking world—for example:
Redditch United axe playing budget as manager leaves [BBC News]
Winnipeg Police are looking for suspects after three men were allegedly attacked with an axe late Friday night. [CBC]
Last month, the district council put forward proposals to axe travel tokens for its residents to save £88,000 a year … [Herald Series]
This Ngram charts the use of ax and axe in American books published from 1800 to 2019:
Though this graph shows ax only slightly ahead, searches of current articles in American news publications shows approximately a 2-1 ratio in favor of ax.