I think the no-lye gets your hair straighter. I used no-lye in college because the bss didn't seem to sell lye relaxers. In college my hair was very very straight. I had to use a ton of products to keep my hair moisturized however because it was very dry.
When I did get a hold of lye products in college (access to a car), I could see a difference in my hair where the lye started and where the no-lye started. My hair was frizzier in some spots because I wasn't relaxing my hair to bone straight. I can't really explain it. Now with my lye relaxers, I can't get my hair straight straight. I loosens the curl and that's about it, but it doesn't look a hot mess when I dry it.
I believe (now this is just me) that with a no-lye you really can't have room for error. If you underprocess your hair isn't going to look as good. You have to be diligent in your relaxing.
I'm not sure about the results of using no-lye over lye, so I would suggest stick with the relaxer you are using.
I think I would use a no-lye relaxer if I wanted my hair to be bone straight or very very close to it. I'm too scared to experiment and try it now. I did back in college, but I didn't really know then what I know now about my hair and I wouldn't want to take that chance.