If I touch the anode of a battery or connect the anode the the cathode of an other battery, none (or rather: a very small current) flows between those two. If I instead connect the connect the anode and cathode of the same battery, a current flows between those two points. Why is this? After all, there's a voltage between me and the anode.
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
A battery works by making charge want to move from one terminal to the other (we call this a potential difference, or voltage). The charge is also moving through the inside of the battery itself (the electrolyte).
When you hook up a wire to an anode and cathode of two different batteries, the buildup of charge on the anode of the first battery quickly "equalizes" with the cathode of the other battery. That's why you'll see a possible short bit of current, and then none. There is no path for charge to continue flowing. There is no common electrolyte between the two batteries for the chemical reaction that generates this voltage to take place.
Think of one battery in short circuit with one wire like a circle (this is gross oversimplification but we will use it as a model). The charge goes through the wire, through the electrolyte, through the wire again, etc. (Doesn't happen quite like that, it actually is based on the creation of a potential difference, but it's hard to model that.)