Hashtags are great, but they're really just a type of index containing a specific type of topic. They have no notion of membership, and the index itself only contains posts that your instance is aware of.
The local instance timeline is useful if you have an instance that revolves around a certain topic or theme, but it's really just for checking out what's going on in your local instance. For large instances, or instances that have more than one topical theme in their community, this isn't useful for group communication.
Where groups ultimately differ is this: basically, it is a subscription relay that announces all posts that are addressed to it. Every member of the group receives every post addressed to the group. This differs from hashtags because you have a chance of missing posts that your instance doesn't know about, and you usually have a member that can administer group membership for moderation purposes.