I forget the full legal arrangement, but I think Twitch at this point owns Curse or at the very least has a hand in managing it.
It's the Twitch Client that exports and imports the modpack files that are posted to and distributed by CurseForge.
Frankly I'm not impressed by the Twitch Client at all, so I would recommend using Twitch to download the pack but not play it, I would instead recommend following this video to run the Twitch pack through the vanilla launcher: