Choice between A13 Z and E13 for a post-doc position in Germany
A E13 position may have a little shorter working hours per week, for eaxample in North Rhine-Westphalia you have to work 39 hrs 50 min/week as "Angestellter" (E13) and 41 hrs/week as Beamter (A13). At least until you are 55 years old, as the working hours for Beamte get gradually reduced as you grow older. But I guess this would not apply to a post-doc position as academic work groups usually do not record the time of their scientist (this means, they often work longer hours without recompensation). 

As posted by Roland in his comment above, a Beamter has to get a private health care plan which can be expensive, depending on your age and health status, but as the state pays approximately half of your health care bills, the conditions Beamte get by the private health care companies are much more favourable in comparison to the ones offered to "regular" clients and in the end a Beamter has a higher net salary.

A Beamter has furthermore to pass a health test and perform a vow on the German Grundgesetz (constitution) and the German laws and is not allowed to go on strike. In case of permanent positions, the pensions for Beamte are significantly higher than the pensions for Angestellte. So if you are planning (or hoping) for a permanent position I think a position as Beamter might be more beneficial.

Your family status is also relevant for your salary as Beamter, this means, if you are married with children you will get a higher salary.