What is the difference between 刚刚 and 刚才 in Mandarin?
Both means the same, but which one sounds more natural depends on what the questioner uses.

if they ask: 

你刚刚去哪了？

you answer with: 

我刚刚回家了。

if they ask: 

你刚才去哪了？ 

you answer with: 

我刚才回家了。

Of course, it's not wrong to answer 刚刚 with 刚才, it's not that big of a deal, both works, just a matter of preference really.