I would think they would - there are different sorts of synthetic hormones, and each of us reacts to each sort differently (that's why we often have to switch around till we find one that our body is happy with). Example: I LOVED Ortho Tri-Cyclen Lo. My skin loved it. My uterus loved it. My hair loved it (it was thick if not defined curl-wise). That has norgestimate as the progestin and ethynl estradiol as the estrogen. Apparently those synthetic hormones get on well with me. I switched to the POP Cerazette when I moved to the UK, which uses desogestrel as its progestin. This did not get on well with me: I had massive mood swings, my hair started falling out and thinned dramatically, and my skin got almost as bad as it was when I was a teenager - and this was after 9 months of using it! I've since switched to a Mirena, which releases an extremely low dose of levonestrogel progestin locally on the uterus. My skin had a bit of a hiccup as I switched methods, but I've had no ill effects I can attribute to the hormone itself (I also used Levonelle before, which has the same progestin at a much higher dose, with no problems).
Certain BC pills are touted to be 'better' for skin or hair (I seem to remember Yasmin and/or Yaz was advertised as the best for skin), but it all comes down to how well your body reacts to it individually.