You know, the more I think about it, I’m not really sure I have any idea what a computer club actually is and does...
Only socializing online is just not the same.
Some hackerspaces are more a kind of makerspace and provide expensive, large or complicated hardware like industrial laser cutters, 3D printers, embroidery machines and (electronics) workshops with soldering irons, electronic parts etc.
Other hackerspaces are focusing more on the social side and offer a space to hang out, meet and discuss with beverages (I guess mostly mate and beer). There can be talks, workshops or competitions (like CTFs) and so on.
Also providing services to the public, like repair cafes and holiday programs for kids can be a way to further engage in society to share technical knowledge.
hackerspaces.org has also extensive explanations on theory of hackerspaces: https://wiki.hackerspaces.org/Theory
That there is also a branch of very public security experts is... Incidental, I'd say.
No. It happens that a lot of members are security experts but it is far far broader than that.
The CCC is a very decentralized organization. A lot of hacker spaces are in fact operated by local subdivisions or are completely independent organizations but with a lot of overlap in membership.
In general the CCC likes define itself more by those who share its values and less by the legal entity with that name.