I am using the translation by Joan Stambaugh. Can someone explain what is meant by "Da-sein", and how does this compares to the more used "Dasein"?
First, one needs to understand how Dasein should be understood in Sein und Zeit (SuZ in what follows). It stems from the German word "Dasein" that can be translated by "existence". "Da" in German means "there" and "sein" means "to be/being". So "Dasein", in its original sense, could be understood as "being there", some kind of euphemism for "existence".
Now, and that is the crucial point, Dasein in Heidegger's work, should not be understood as "being there". Here is an excerpt from a letter written by Heidegger and addressed to the French philosopher Jean Beaufret (Questions III et IV, p. 130 — I unfortunately couldn't find the original letter so this is a translation in English from the translation in French from the letter in German):
"Da-sein" is a key word of my work, and it is victim of serious misinterpretations. "Da-sein" does not really mean for me "here I am !" but, if i may express myself in a likely impossible french: being-the-there (être-le-là) and the-there (le-là) is precisely Aléthéia, disclosure — opening.
It appears clearly that the "Da" in Dasein has nothing to do with the usual spatial "there" but must be understood as the opening in which the world is projected. And this explain why this "Da" must be as such constituted (see SuZ, V, "The existential constitution of the There", §§29-34.)
From there, let us see how this helps for the distinction of the terms Dasein and Da-sein. Take the following excerpt (SuZ, § 28):
Der Ausdruck »Da« meint diese wesenhafte Erschlossenheit. Durch sie ist dieses Seiende (das Dasein) in eins mit dem Da-sein von Welt für es selbst »da«
The Stambaugh translation reads (removing the hyphen that they mistakenly added inside the parenthesis):
The expression "There" means this essential disclosedness. Through disclosedness this being (Dasein [and not Da-sein]) is "there" for itself together with the Da-sein of the world.
Here, we see the contrast of the two words. On one hand, Dasein means this being (Seinde/étant) "who is concerned in his Being (Sein/être) with his Being (Sein/être)" (»Diesem Seinden in seinem Sein um dieses Sein selbst geht«, SuZ, §16.) On the other hand, Da-sein is now the Da-sein of the world. What does it mean? We could translate it further as "the Being-the-there of the world", that is being the opening (Da) open for the world to be projected in. And this way for Dasein to be-in-the-world ("In-der-Welt-Sein") is actually the constitution of the Dasein (see SuZ, II.) ; which is the reason why the Dasein as being (Seinde) is one ("eins") with the act (or the fact) of Being-the-there of/for the world. And in this union from Dasein and Da-sein, the Dasein itself must be the Da for himself.
As always, it gets very finicky and kind of a perpetual self-referencing. This is in some extent the direct consequence of the definition of Dasein : "This being who is concerned in his Being (Sein/être) with his Being (Sein/être)". There is already in this definition a loop that Heidegger keeps on unrolling through SuZ.
But from there emerges the difference and the profound relation between Dasein and Da-sein. They are two side of a same thing: Dasein is the being (Seinde) that Heidegger is interested which constitutes itself in the act of Being-the-there (Da-sein) for the world it projects and in which it projects himself. As constituting this act or fact is one with the being. (Note that I use there the term "act" in a loose way and it does not imply anything like intentionality, conscience, ego etc. On the contrary, the constitution of the Dasein precedes any of those.)
I answer here my own question as I have found it in the book of Thomas Sheehan, Making sense of Heidegger, 1st ed. Rowan & Littlefield Int. (2015). On p135:
Heidegger's term ofr any concrete, personal instance of hermeneutical openedness is Dasein, whereas his precise word for the "essence" or ontological structure of any concrete, personal ex-sistence is Existenz or Da-sein (usually hyphenated, but Heidegger is not always consistent).
If anyone is checking this reference, I encourage to (sic!) write the book of Sheehan in order to grasp how he uses all the terms such as "essence", "ex-sistence" etc.