Bias gives the quality of stretchyness. This is very useful if you are a wee bit short on fabric for your binding, such as if you make a slight mistake in measurements. The bias allows you to coax, for example, a 3 ft piece of binding to ft an edge 3 ft 3 inches in length.
The stretchyness allows you to bind around curved corners, scalloped edges, or a round quilt (mug rug or table runner) easily and smoothly.
Matching patterns: if your binding has a printed recognizable pattern, cutting on the bias allows short pieces to be matched and connected with less attention to whether or not that print is recognizable. On a small piece, a too-bold, too distinct pattern on your binding could be a distraction.
These are my reasons. I've used straight-grain binding with both success and frustration. When I'm working on a very stable, perfectly squared quilt, then straight grain binding is okay.