My answer:

Hi, there’s no big difference between 才 & 刚 when both of them indicate the thing just happened not long ago. For example, 我才/刚起床。 You can even say 我才刚刚起床。 But 才 has a function which 刚 doesn’t have – 才 can indicate that something happened or was done much later than expected or what it was supposed to be. For example, 我们说好了九点见面，你怎么才来啊？（let’s suppose it’s already 10 o’clock） You came much later than the agreed time! 

Perhaps you’ll have to distinguish 刚才 & 刚刚. 刚才 is a noun of time, while 刚（刚）is an adverb of time. 我刚才/刚刚去了超市。Both 刚才 & 刚刚 are ok. I went to the supermarket just now. 刚才的电话是谁打给你的？ Only 刚才 can be used here. (because the structure is __ + 的 + noun) Who called you just now? (the phone call you received just now is from whom)