Julian Assange hasn't been charged with 'publishing classified information' either. In both Australia and the US you can be prosecuted for participating in another's crime of leaking classified information. The question is, what qualifies as participation (aiding, conspiracy, etc), and is there an express or implied exception for the activities of the publisher in question?

The police raids, which are still too newly performed to be definitive about, appear to be hunting for the original sources. No charges have been leveled against the journalists involved, yet.

It took them years to suddenly care to hunt for these sources, so the whole situation is... Odd.

What would be difference between sharing and publishing, from a legal perspective?

I think usually the idea is you prosecute the person who steals the information, but if they pass it to another person who publishes, that person can't be prosecuted.

If the same person stole and published I'd expect they could still be prosecuted.

But others resharing it, (eg if someone posted a pastebin and others shared) would not.

In the US the distinction is made so that journalists exposing misdeeds from people in power cannot be prosecuted since the value of the information is too high for the public.