A battery is no capacitor, and the actual charge stored in the battery terminals is very low. When you connect the anode of one battery to the cathode of another, that charge is transferred very quickly, and the voltage drops to zero.
When you connect anode and cathode of the same battery, a chemical reaction takes place, and charges flow inside the battery as well. For example, in a lead-acid battery, $H^+$ ions and electrons are produced at the anode. They have a total charge of zero. The electrons move through the (external) wire to the cathode, while the ions move through the electrolyte inside the battery to the anode.
If you connect anode and cathode of two different batteries, the ion current does not flow, and the imbalance builds up a voltage that neutralises the initial voltage between the two batteries. Thus, the current stops.
The full reactions are (see http://en.wikipedia.org/wiki/Lead–acid_battery):
Anode: Pb + HSO$_4^- \rightarrow $ PbSO$_4$ + H$^+$ +2e$^-$
Cathode: PbO$_2$+HSO$_4^-$ + 3H$^+$ + 2e$^- \rightarrow $ PbSO$_4$ +2 H$_2$O