Hashtags are passive filters for your server's federated timeline. They will show posts from other servers, but only if your server has noticed the post anyway (there's a list here of what servers notice: https://fedi.tips/which-posts-and-accounts-can-i-see-from-my-server/).
Groups work totally differently, they actively push all their content to everyone who follows the group, even if their server hasn't noticed that content yet.
So, Groups federate much more effectively.
Hashtags are passive filters for all the content that your server has noticed (if you're interested, there's a list of what servers notice at https://fedi.tips/which-posts-and-accounts-can-i-see-from-my-server/).
Groups are actively pushing all of their content to all the people who follow the group, no matter which server they are on.
So, they seem similar on the surface in how they work, but underneath the surface groups are a lot better at distributing posts.
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