I guess it's a historical vestige from the times chassidim were all about wearing silk and not wearing woolen garments in order to avoid shaatnez.
For example, the Chassidim famously go about wearing silk garments. Chassidim began to wear silk because of their meticulousness about not wearing woolen clothing at all, because of a concern that a linen thread may have been sewn into the wool garment, thereby prohibiting it as sha’atnez. The Chida, living in Eretz Israel, was a witnesses to the beginning of the Aliya of the Chassidim, and he asked them about the practice of wearing silk. He reports the story in his flowery language:
Years ago a band of scribes came from a foreign land to settle in Eretz Israel. Two of them were Rabbis who sanctified and purified themselves in a meticulous manner. All could see that they dressed themselves from head to toe in silk, and I asked these Rabbis to explain the practice… The Rabbis said that in their opinion linen threads are woven into all the woolen garments in the world, and they are sha’atnez. Therefore they treat all woolen garments in all places as sha’atnez. This is what they said. From that point on a fire burned within me, but not one of the Rabbis or pious men of our holy city was concerned about this, for in all corners of the world Jews wear woolen garments from all places as is well-known. (Birkei Yosef, Yoreh De'a 299)
Why the corners are of different fabric than the main fabric of the tallis? Why not be an all woolen tallis like the sefaradim prefer?
Because the main fabric is made entirely woven (at least historically), it isn't cut from a bigger piece of fabric so you don't need to sew a hem to avoid unraveling. Being entirely woven and also being imported from a place where it was forbidden to weave wool and linen together (the turkish tallis) then you would be safer knowing that you were not wearing shaatnez and still performing the mitzvah in the best way.
The corners are definitely cut from a bigger piece of fabric. At that time the dominant fabrics especially in eastern Europe were wool and linen, no synthetics and silk and even cotton were luxury items from distant lands. People would'nt buy another imported safe tallis-sized fabric just to cut 4 small squares from it, so to avoid shaatnez from the corners they would just buy a piece of fine silk (which isn't mixed with wool or linen and even if it were you could visibly recognize it) at the local store, the thread used to sew would also be silk.
With the invention of synthetic fibers, their dominance along with cotton, and modern methods to identify fibers (e.g. microscopy) this isn't an issue anymore to chassidim and survives just as tradition.