～組 usually follows a founder's family name (e.g., 山口組, 東組, 浅野組). This is because ～組 was originally a naming convention of Japanese carpenters, who worked in a smaller, family-like organization in the past. There are still many (benign) general contractors named like
family-name + 組 in Japan (see this list). Japanese yakuza borrowed their convention.
～会 is a more generic suffix which can follow a place name (e.g., 住吉会 named after Sumiyoshi Town, Tokyo), a "concept" (e.g., 共政会 "cooperatively governed", 旭琉會 "rising sun of Okinawa", 親和会 "fellowship") as well as a person name (e.g., 工藤会 founded by Genji Kudo). There are tons of NPOs named like ～会.
Anyway, once it has been established as the name of an organization, there is no meaningful difference.