This is one of the biggest myths. Of course, it makes a difference - if you don't have a ligature on your mouthpiece it will not sound the same as when you do have one.
Seriously though, the role of a ligature is purely to hold the reed securely onto the mouthpiece, so the only way that it makes a difference is whether it is doing that properly or not. If the ligature is broken, not working correctly or designed for a different mouthpiece then it will not be working as it should and so it may of course sound different - but only because it isn't fitting well, not because the ligature per se has a different sound. Sometimes a mouthpiece can have a concave or convex table and in that case certain ligatures may not hold the reed so well and the same applies. However there is no such thing as a ligatures that sounds better than another one.