I personally love needleturn. I have the most control of the outcome with that because I am manipulating the fabric as I sew. I also love that the stitches are virtually invisible & that I am working with nothing but cotton fabric; it produces a very soft quilt. However, it is one of the longest, slowest methods for applique.
For a quick quilt, I like fusible (with windowing). I also use fusible with wool/felt applique; I think it looks adorable & it is the only way other than if I base the felt applique down before stitching, that I can work with felt. I've done fusible with cotton as well. It does cause the edge to fray slightly (how much it frays is determined by the length/width of your stitch; the longer & wider your stitch, the more it will fray), but with a matching thread I can get perfectly crisp corners & it's still hard to see from more than a couple feet away.
I've also done the freezer paper method. It does produce crisp edges, but it's personally my least favorite method. You get the results of needleturn without having to mark your fabric. I don't mind marking my designs on my fabric (background & applique pieces). In addition, when I have a tricky piece -- like a circle -- I use mylar template plastic to press it. It's reusable (either as is, or cut down to a different shape) and is not as prone to causing stretching on the edges as what can happen if you remove the freezer paper before stitching (some people leave the freezer paper in their quilt; some cut through the background fabric to remove it after stitching; still others remove it as soon as the piece is pressed & had a chance to cool to room temperature).
There really isn't a right or wrong way to stitch anything. There are award winning quilts that were stitched raw edge with a straight stitch. It really is up to your personal taste & preferences as to which one(s) you use when. I would encourage you to experiment with the different types & see which one(s) you like.