The two words mean the same thing and can be used in the same situations, but one is preferred in American English and the other is primarily used in British English.
Ax without e are the spellings used in American English.
Examples
Youth receiving centers face budget ax
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.
Axe is standard in varieties of English from outside the U.S. and is common in British, Ireland, Australia and Canada.
Examples:
Redditch United axe playing budget as manager leaves
Winnipeg Police are looking for suspects after three men were allegedly attacked with an axe late Friday night.
