There's one other important reason that I didn't see mentioned. When you just cut the square in half and sew it, your square will be smaller than you want it to be. Half-square triangle squares are 7/8" bigger to compensate for the seam allowance that you take out of the middle. There are some great tools out that help with these. I like to use Thangles if I need more than 4 triangles (2 squares). You cut strips of fabric the measurement of one side, then lay the Thangle on top. It is paper and has all the stitching and cutting lines clearly marked. Once you've sewn all the lines, then you cut them apart and the paper tears off. They come in sizes 1 1/2" on up so you just buy the size you need for the finished square, and Presto! they are all cut correctly. There are very good instructions with each package and a video on Thangles.com.
I'm working on a quilt that needs many of these triangle squares and have done it a little differently. I don't cut the fabric strips, just lay the Thangles side by side on the uncut fabric to get the number I need. It depends on the size, but I can get 10 2" finished squares from one paper strip. I tape them together in a couple of places, then pin to fabric. By lining up the stitching lines you can sew 40 squares with just 4 Thangles and long rows of stitching. Doing it this way is a real time saver. Don't we all need that!
:D :-D