Generally, the two can be used interchangeably. As those before me have said, the words 'luck' and 'fortune' are synonymous, although they are not identical.
'Luck' leans more towards chance. It is more chaotic. You are lucky when the outcomes of a randomly decided event are more often in your favor then they are not.
'Fortune' tends more to imply an external force, such as divine assistance, that is outside of your realm of control. This could possibly be attributed to the fact the we get the word 'Fortune' from 'Fortuna', the Latin name of the Greek god Tyche who was a god of fate. He was in charge of determining whether people had good or bad luck.
Of course, both words, like most, also have there secondary meanings that stray from the topic, such as fortune also meaning wealth.
Luck is more a word statistical well-faring, while Fortune caries implications of a divine hand at work.
Unfortunately, this distinction has faded in the rapidly decaying state of our societies literary competance. :(