Your question mentions both activities and job offers, for which the situation is slightly different. This answer is specific to Germany, and from the perspective of Mathematics. (The formal structure is the same in other disciplines, but how it is handled in practice can differ significantly.)
Regarding activities: What a habilitation confers is the so-called venia docendi, the right to teach at the university.
The two main consequences are
The right to give lectures at the department.
The right to supervise PhD students in your own name (i.e., not just be co-supervisor together with some other full professor).
While the first line is somewhat blurred in practice (for example, the department can give you a teaching assignment for a lecture even without habilitation; on the other hand, the right to choose your lectures to teach is somewhat less useful if the department decides that those lectures will not count toward your teaching quota, so it would have to be on top of your regular duty of four classes per week), the second to my knowledge is hard.
These rights are automatically conferred to full professors upon instatement, so the practical role of habilitation is a sort of promotion inside a department from postdoc to assistant professor (a role that is supposed to be supplanted by the junior professorship). There is also a formal procedure for transferring your habilitation from one (German) university to another. I believe it's also possible to get a foreign habilitation acknowledged and transferred, but that is more involved.
The rights and requirements for a habilitation and the rules for the procedures are university or department policy.
It also plays a role in appointments for a full professor position.
In Germany, the relevant requirement in job offers is usually worded as "habilitation or equivalent qualification". The latter option is routinely used for foreign candidates or younger researchers they want to fast-track. If this is done, it is part of the duty of the hiring committee (and in case of candidates that make it to the short list, the external reviewers) to explicitly state which contributions were considered as equivalent qualification. This could be a direct foreign equivalent (such as the French habilitation or the Russian Doctorate), a series of high-impact papers, a published monograph (or particularly nice textbook) or successful direction of PhD students. The requirement is also routinely waived for candidates already holding a tenured position (including junior professors after a positive evaluation), the "equivalent qualification" presumably being having received tenure in the first place. As I wrote above, instatement as a full professor automatically confers the venia docendi, so formally this makes sense (at least within the German system.) The legal framework here is somewhat more difficult; schools and universities are the responsibilities of the state, not of the federal government, and the autonomy granted to the universities differs from state to state. In general, the requirements for a full professor position are set down in state law ("Landeshochschulgesetz", usually in the exact form I stated). The decision of the hiring committee, including the recognition of equivalent qualifications, has to be passed on to and sanctioned by the department, university board and (depending on the state) the ministry.
