Funny you should write "brillo pad" as I just wrote a post about this on another forum. Okay, long response, hope you all don't mind.
Some things that are working for me:
1. I think it's difficult for well moisturzed natural hair to feel like a brillo pad. That first inch or so of new growth is the worst. It's not soft or pliable at all. As Brittanynic mentioned, it is probably residual effects of the relaxer. I didn't have that much damage, but even still, my new growth was rough and dry (not thick or nappy, which is different than dry and brillo pad feeling) until about the third month. I also discovered I wasn't moisturizing enough (see below). My "real" texture didn't start coming in until toward the end of the fourth month of transitioning.
2. I have discovered that my natural hair needs a lot more moisture than my relaxed hair does. Even now, when I wash my hair at night and air dry it with leave-in, my natural hair is almost dry by morning. My relaxed hair is still very wet. I probably moisturize my hair 60% more now than I did as a fully relaxed head. When my hair is too dry, some parts do start to feel like a brillo pad. When it's properly moisturized--all the textures--napps, coils, spirals, and frizz--are soft. My hair doesn't seem to love oils much now--it needs water based moisturizers, but that might change.
3. I have discovered that generally, my hair does not like combs very much. My hair is super tangly--meaning I can comb through it, and then go back over the same section and there are tangles again--I think I have lots of O shaped strands. My hair responds much better to finger combing, where I can control the tension better, with an occasional wide-tooth comb or Denman brush when my hair is wet or damp to capture shed hair (which is a big source of my tangles). Just this week I went with the recommendation of folks who detangle under water, and I will tell you my detangling session went a lot better, so now this is a part of my new detangling routine. I also wash my hair in at least four sections and detangle each section separately.
Always keep in mind that most of us have had relaxed hair for 10, 15, 20 years, and even then it probably took most of us years to learn how to properly care for our relaxed hair. It will take you a while to figure out how to take care of your natural hair too--I'm only 6 months post, and everyday, I learn something new about my hair. Things don't aways go well for me during this transition, but it's worth it--I look at my natural hair, and I sometimes feel a little shame that I hid it for so long.