The Ashkenazic form of windings are 7, 8, 11 and 13. What is the significance for these numbers? The Sephardic form of windings are 10, 5, 6 and 5. What is the significance for these numbers?
The Sepharadi part of your question is straightforward. If you translate those winding numbers -- 10 + 5 + 6 + 5 -- back though gematria, you get Hashem's name: י - 10, ה - 5, ו - 6, ה- 5.
The Ashkenazi side is more nuanced. As sam wrote in his answer, the number of windings is 39, which is to equal Hashem Echad (Hashem is One - Unique and Indivisible).
The Beis Yoseif says the 5 knots are so that when we hold the two front corners while saying Shema (yes, I know there is a different custom to hold all 4 corners), we have 10 knots, corresponding to the 10 sefiros through which Hashem's Presence is felt in creation.
In the gemara's system, there are 7 to 13 groups of windings. And it sounds like each group was 3 windings. (Majority opinion, Rav Natrunai Gaon reads it as "once, twice, three times, [and so on]".) Rashi, Tosafos and the Moredechai say that the gemara is specific to winding tzitzis with techeiles, and not applicable to us. But either way, it shows there is some kind of significance to 39 windings because it's 3 * 13. No "why" there, just an indication the number is significant to the mitzvah of tzitzis.
But 39 also comes up in at least two other contexts: the maximum number of lashes the court may give as corporal punishment (if the person is healthy like an ox, and could take the maximum), as well as the number of melakhos, categories of constructive labor, prohibited on Shabbos.
Lashes are meted out in groups of three, separated by doctor inspections.
7, 8, 11, 13, though, closely matches the way the 39 prohibited activities of Shabbos divide up. The first 11 melakhos (Shabbos 7:2) describe the steps necessary to grow wheat, turn it into flour, and make the showbread. The next 13 are about preparing the cloth of the curtains of the Mishkan, from the wool to the dying to the weaving and sowing. Seven melakhos relate to preparing hides into leather, and the last 8 are simply “none of the above."
As I wrote in another answer recently, Rabbi Aryeh Kaplan explains how 40 represents creation, and thus 39 is the number that indicates human efforts to imitate Divine Creation. (The 40th being creation ex nihilo, something from nothing, which people are incapable of doing.) And so, the person who acts to undermine the purpose of creation gets 39 lashes, and to commemorate resting from creation, we rest from 39 acts.
Ashkenazi tzitzis therefore also invokes creation; having the same 39 divided into the same groups as the melakhos cannot be coincidental.
As Rav Hirsch put it, tzitzis are literally sprouts. They represent the human creative extension of our basic functional garments. They start off bound, channeled by the Torah, after which they are given free extension -- we should express our individuality within the framework the Torah gives us.
The reminder of creation thus fits Rav Hirsch's general approach to the mitzvah.