I'm currently GMing a game in a home variation of Runner Haven's Hamburg and I was wondering how the powers were divided between the AGS and the Free and Hanseatic City of Hamburg. Is it different from today's Germany? And what would Hamburg get if it were to separate from the AGS (and NEEC)?

"Deutschland in den Schatten 2" (Germany in the Shadows 2) outlines the political structure of the AGS, although it's 10 years behind the current timeline, being a SR3 supplement. AFAIK the SR4 timeline didn't significantly change the balance of power (although the AGS is a lot more of a corporate marionette nowadays).
The short version is that germany degenerated back into a loose federation of heavily decentralized states reminiscend of the holy roman empire (SR history never won an award for originality), meaning that apart from a few federally held rights, mostly those concerning the well being of the AGS as a whole (foreign diplomacy, military, international trade rights, etc.), Hamburg is free to do wantever it wants to.

If seperated, they'd be free from federal taxes by the AGS and the trade obligations by the NEEC, which would have some positive economic effects, although every corp or country could theoretically waltz in and annex them, seeing as they'd be independent and having just pissed off the powers most interested in their wellbeing.

The AGS are still responsible for protecting the border, Hamburg is still be responsible for dealing with crime. The later function is outsourced to HanseSec, the former taken by the Bundesgrenzschutz. The "diversification" of the states ensures that the AGS is not capable of acting meaningful on most of the common interests it is responsible for. Think todays European Community.

As for leaving the AGS, this is nowadays a theoretical right of the free states (Bavaria, Saxony), too (IIRC). The NEEC and AGS could level trade tariffs against an outer entity, as Hamburg would have to join any international organisation on it´s own. "Treason" can end ugly.