I've seen some people (notably Sefardim) who sell "soft" Shmura Matzos (they are flimsy like a wrap).
What is the difference in how they are made?
See this handy Hebrew/English Shulchan Aruch, Orach Chaim 460
Matzah can be any unleavened bread, up to a tefach thick.
From the time that the flour gets wet, until the final product is baked, one has 18 minutes to bake the matzah. (see Shulchan Aruch, Orach Chaim 442)
Some sefardim have a custom to make a matzah that is soft, like a wrap. It is a little thicker than hard ashkenazi cracker-matzah (but still much less than a tefach).
Sephardic Matzahs use a shorter 15 minute cycle, because these matzahs take up to 2 minutes to bake, compared to Ashkenazi Matzah that bakes only 15-20 seconds in the oven. From here
Note the Rema's comment on 460:4:
"Rema: The matzos should be crackers, and not thick like other bread, because crackers do not rise quickly."
Because of that, Ashkenazi practice is to make matzos like the hard crackers to which many of us are accustomed.
However, see Rabbi Hershel Shachter's opinon, which allows Ashkenazim to eat Sefardi matzah, because "rekikin" doesn't mean cracker specifically, but simply a matzah much thinner than the halachic maximum of tefach.
It should be noted that, even among the majority of Ashkenazi poskim who prefer that Ashkenazim maintain their cracker-matzah minhag, all agree that sefardi matzah is kosher l'pesach, and certainly there is no halachic violation of any kind for an Ashkenazi to eat Sefardi matzah on Pesach.