Some people think it makes a difference others don't, although it's usually agreed that if it's broken or not working properly then it has an adverse affect on the sound.
My own opinion (and I'm open to being proven wrong), is that if a ligature is working properly, ie holding the reed firmly against a flat mouthpiece table, then they all sound the same.
Some do a better job than others if the mouthpiece is distorted in any way, so this could explain differences in sound.
It's also quite possible that a ligature can have a certain sound very close up that only the player hears, or it might seem to have a different response that only the player feels.
I have tried quite a few, including normal 2 screw, Rovners of all types, Vandoren Optimum and Francois Louis, and to be honest I can't tell the difference. I use either a Rovner or a standard Vandoren 2 screw. I can switch between them even when recording the same song and there is no recorded difference in sound (that I can hear).