It depends what context you are referring to:

Generally:

Good - often referring to the opposite of evil, the quality of an item, or the support of a statement.

Nice - often referring to the characteristic of being kind to other people, the quality of an item, or the support of a statement.

Here are some examples where they CAN mean different things:

Rescuing people from thugs makes you a good person. Murderers are evil. (Rescuing is good, because it represent values that we consider "good." Rescuing people in the face of danger presents a moral "goodness." Standing up for others is good, but if you're referring to the act of kindness, it would be nice. When you're thanked, people would often say: It was nice/kind of you to do that.)

He was a nice boy (respectful or kind or good-looking). He was a good boy (he behaved himself, had good intentions). These often include the other word as well. If you say someone was nice, they might also be good. If you say someone was good, they might also be nice. They definitely overlap.

That is a really nice coat (it looks nice, or is of fine quality or brand - stylish).

It is a really good coat (It works well to keep you warm/dry. Good quality for usefulness)

I won the lottery. Nice, I'm happy for you.

I won the lottery. Good for you. (usually sarcasm, but can be positive)

Examples where they are essentially the same:

You look good/nice today.

That was a really nice/good thing you did to help that person. 

(When replying to someone else) That's nice/good to hear.

I punched that bully at school. Good/nice, he deserved it.

These are just some ways to use the words, but generally they imply the same thing: That something is of good quality, moral character, or when supporting a statement. Hope this helps! :)

In most situations, there's practically no difference. You can interchange them, but you need some insight into the context of what is being said and a good intuition in colloquial English to actually differentiate and determine which one is better to use in any given context.

Thanks,

Mir Tunio