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.