Here's a simplistic explanation.
Usenet is like a distributed bulletin board system. Messages are stored in news servers worldwide, and each server hosts several newsgroups, which are discussion groups devoted to a special topic. Users post and fetch messages to newsgroups using news readers. Messages are exchanged between users and news servers - and distributed from a news server to another - via the NNTP protocol.
Usenet has no centralized authority just like the World Wide Web has no centralized authority; single webservers and websites may be administered by different organizations, but nobody controls the Web as a whole. Just like the Web is the worldwide network of webservers accessed via HTTP/HTTPS, Usenet is the worldwide network of news servers accessed via NNTP.
On the other hand, a forum is an online discussion site implemented on a specific webserver using a specific software (see here for a list). As such, each forum has its own admins and users.