Let's say your friend tells you they are about to take a driving test, go up to talk to a cute girl/boy at a bar, go to hospital for an operation, or climb up a cliff face. What do you say to them?
In English, most people automatically say "Good luck!"
In French you probably say "Bonne chance!" which means the same thing; in German it's "Viel Glück" (same again), or "Mach's Gut!" (make good!)
In Japanese the standard no-thinking-about-it phrase is 頑張って！ がんばって！ ganbatte! (or gambatte!) (alternatives: 頑張れ！ 頑張れよ！頑張ってね！ 頑張って下さい！）
In the west (at least in the languages I'm familiar with) you wish the person to have luck, which is a something that is bestowed upon them by the random forces of the cosmos. I have a feeling, though I may be off on this, that wishing someone luck evolved from the religious equivalents "May God be with you!" "My prayers are with you!"
Of course there are ways to say Good Luck in Japanese, but when you wish someone luck you are consciously choosing to say so. For instance you might say to someone who's bought a lottery ticket, ご幸運をお祈りします - ごこううんをおいのりします gokouun o oinorishimasu. I wish you (pray for your) good luck.
But the phrase that is automatically said in most situations has nothing to do with luck. The base form of the Japanese phrase is 頑張る がんばる ganbaru (gambaru) which means to strive, to try ones best. So 頑張って means 'Go and try your best!' The onus is put on the efforts of the person or persons involved in the endeavor, rather than random forces or a higher entity.
A subtle yes interesting difference, no?
(The German Mach's Gut might be closer in intent to 頑張って. Not sure about the theatrical Break a leg! - does that mean that the actor tries so hard that he's supposed to break a leg, or just reserve-luckology, the equivalent of wishing someone bad luck?)