Biobatteries function on similar principles to many animal metabolisms; compounds are broken down via enzymes, releasing energy in the form of heat and charged particles. Those charged particles are then distributed to the cathode and anode of the battery, creating an electrical potential. In more conventional terms, you could put food in, and get electricity out.
An advantage is that this battery can be charged with a much wider array of fuels, rather than just electricity. This allows us to harness energy that would have otherwise been entropically lost.
One obvious disadvantage is that the food will not be available for any organism to eat. This is on par with the debate about ethanol (whether to use corn for fuel or food), however my current understanding of world food supply is that we have more than enough food for everyone, it's just not equally distributed and we waste a shocking amount of it (many restaurants throw out multiple pounds of edible food every day, for example). Additionally, these products would not be available for other non-food uses, such as fertilizer, animal feed or biofuels. Thus, biobatteries may be best suited for conditional uses such as public sanitation or in-home garbage disposal.
Additionally, they would need to be supplied with a steady stream of enzymes in order for the reactions to take place at all. This might pose some additional complications in terms of design and costs.
There are also potential issues with contamination, such as the wrong bacteria getting into the battery and rendering it hazardous to anyone who touches it.
Finally, a very significant disadvantage is the actual yield of the battery; simply having enzymes present will not fully utilize the energy in the glucose to the degree that a living cellular organism could. This will render the battery relatively inefficient. For example, if glucose is broken down to GDL (c6h10o6) this generates only 2 protons and 2 electrons; a reasonable amount of energy compared to a conventional battery, except that the glucose and GDL molecules are significantly larger than the monatomic or diatomic molecules used in conventional batteries, not to mention the size of the enzymes, resulting in a much lower overall energy density.