Lye gives more bounce IMO. I'm fairly new around these parts but you may want to do a search for a similar topic. Unknown it's been discussed on here somewhere....getting the swang from lye vs non lye.
My lye relaxed hair was more bouncy and sleek then my no-lye relaxed hair.
No lye meant the ability to self relax without depending on a stylist but also more stiff hair that would not stay sleek.
I tried Vitale as my first lye relaxer and I was happy, but once that ran out I tried and fell in love with Linange Shea Butter. It seems to agree with my fine strands.
I used no lye relaxers when I was a child-young teen and they didn't compare to when I switched to lye. My hair had a silkier texture, was more moisturized. It had body and movement.
Stick with lye. Lye doesn't make some people's hair as straight but that's why the hair isn't so stiff. It keeps some of the hair's natural texture and elasticity. Elasticity is what allows your hair not to break while it's being groomed.
I've only used just for me, then switched to Muzak butter blends and now I use a lye relaxer. My hair was always silky and bouncy and didn't change depending on the relaxer or relaxer type