What is the difference between non-self and emptiness?

Anatta:

Is the reality that we cannot control the nature of Anicca and the result of it(Dkkha).

Sunya:

That mean nothing could gain from it. That hasnt any advantage towards the way of Nibbana.

1. What is the difference between them?

As far as I know, there is no difference. The world is said to be empty of self and what belongs to self.

Then the Venerable Ānanda approached the Blessed One … and said to him: “Venerable sir, it is said, ‘Empty is the world, empty is the world.’ In what way, venerable sir, is it said, ‘Empty is the world’?”

“It is, Ānanda, because it is empty of self and of what belongs to self that it is said, ‘Empty is the world.’ And what is empty of self and of what belongs to self? The eye, Ānanda, is empty of self and of what belongs to self. Forms are empty of self and of what belongs to self. Eye-consciousness is empty of self and of what belongs to self. Eye-contact is empty of self and of what belongs to self…. Whatever feeling arises with mind-contact as condition—whether pleasant or painful or neither-painful-nor-pleasant—that too is empty of self and of what belongs to self.

“It is, Ānanda, because it is empty of self and of what belongs to self that it is said, ‘Empty is the world.’”
-SN 35.85, World Emptiness (Suñña­ta­loka­-sutta)

Mogharaja: Twice now, O Sakyan, I've asked you, but you, O One with Eyes, haven't answered me. "When asked the third time, the divine seer answers": so I have heard. This world, the next world, the Brahma world with its devas: I don't know how they're viewed by the glorious Gotama. So to the one who has seen to the far extreme, I've come with a question: One who regards the world in what way isn't seen by Death's King?

The Buddha: Always mindful, Mogharaja, regard the world as empty, having removed any view in terms of self. This way one is above and beyond death. One who regards the world in this way isn't seen by Death's King.
-Sn 5.15, Mogharaja's Question (Mogharāja-māṇava-pucchā)

This is not the only meaning of emptiness in the early texts though.

They are not the same. But Suñña is one aspect of Anatta. Suñña can mean devoid of any inherent value, goodness or a soul. Anatta means not under one's control. Something that is Anatta isn't always Suñña. ex: Nibbana. Nibbana is peaceful, devoid of suffering and permanent. So there's goodness to it. But it's still not a self or a soul since it's not under your control.

There are 40 meditative objects described in Patisambidhamagga under Vipassana meditation. Ten kinds from the view-point of anicca; twenty-five kinds from the view-point of dukkha; five kinds from the view-point of anatta. Suñña is one of those 40. The detailed explanations are in Visuddhimagga.

I'm new here, thus have not acquired enough points to add my comments above, so I'll do so here. But I agree with the thrust.

I'll sharpen my answer by saying right off the top there is no difference between non-self and emptiness. They are one and the same thing. To comprehend emptiness you will already comprehend non-self, just different ways of looking at reality. Different vantage points.

Emptiness is the key concept in Buddhism. It lies behind everything in Buddhism. If a thing is empty it has no permanent substance. Why? Because it is always in transition, is in the midst of change, is not immutable. If it is in constant change it cannot satisfy, is therefore dukkha, suffering. We the observer are also in the midst of constant change, and therefore share the same characteristic this discussion began with — we are also empty. (Despite all appearances to the contrary.)

These three are the three characteristics of all phenomena. Everything is not only subject to change, but that change is constant. Anicca. Everything ultimately cannot satisfy. Dukkha. There is no permanent reference point, no self. Anatta.

The questioner makes an almost inevitable error. He/she assumes the puzzle of emptiness can be understood analytically, rationally. That by breaking it down into its parts and coming to understand each in turn, assemble a comprehensive understanding. It can't be done.

The only way is to confront it all at once, as a whole. Proceed by looking at what changes. I say that includes anything and everything. Am I right or am I wrong? Can you identify anything that is lasting? Perhaps a rock or a mountain. But these change as well, although not rapidly. Even a mountain is rising and/or falling.

Everywhere you turn there is impermanence. Everything is in process. Everything IS process.

I'll respond in your comments to the above.

When we look for "impermanence" we find only a single instant - hence "impermanence" (the process of change from moment to moment in the form of a "continuity" comprised of past, present, and future) is not there. (There is no continuity as there is only a single instant).

When, within that single instant, we look for a thing, we find only non-thing parts - hence, that so-called "thing" is not there.

And when we look at those parts referred to as "self" in the theory of self, we find that each of them is not self - hence, that so-called "self" is not there.

And since none of those parts is "self", neither can any part be owned by a self - because there is no self there which could own any part.

I would add to this, that in the reception of these two terms, there is a difference. While the two terms are possibly used synonymously in the second sermon of the Buddha (compare the answers already given for this), the terminus technicus in early buddhism is anattā/anātman, being as one of the three marks of existence a claim of the Buddha which marks the uncoupling of Buddhism from the philosophical mainstream of that time.

The word(s) suñña/śūnya have acquired a more refined philosophical sense, since they were to a great extent 'popularized' by Nagarjuna and the Śūnyatāvāda school of early Mahayana. In his main work Mūlamadhyamakakārikā Nagarjuna does in a (philosophically) much more thorough way than was done by the Buddha dissect certain supposed elements of existence, proving, that they are empty. By and large this is in response to the rival buddhist ontologies (Sarvastivada and Sautrantika) and in the end more or less re-states what the Buddha had already stated: anattā/anātman, but on a much more sound philosophical and argumentative basis. (By the way: it reiterates also the Buddha's contempt for metaphysics)

In short: In discourse, anattā/anātman makes one think of Buddhism and the Buddha, while suñña/śūnya incites more special associations to Nagarjuna and his school.

Please note, that the adjectives used in comparison of the words of the Buddha and Nagarjuna in no way constitute value-judgements. They just indicate, that Nagarjuna responded to different needs and not, that what he said is worth more.

As I understand, Mahayana's shunyata (noun, because shunya is adjective) includes all Three Characteristics of Existence, as well as Nirvana (sometimes counted as the fourth!)

In this sense, Shunyata is the all-encompassing view, and as close as it gets to Ultimate Truth, while Anatta is but one aspect of this truth.