There are many advantages of this design over the classic lithium ion design. Because of the solid polymer electrolyte there is no need for the organic flammable solvent. Thus these batteries are less hazardous if mistreated. Furthermore, since no metal battery cell casing is needed, the battery can be lighter and it can be specifically shaped to fit the device it will power. Because of the denser packaging without the holes between cylindrical cells and the lack of metal casing, the energy density of Li polymer batteries is over 20% higher than that of a classical Li ion battery.
The Lithium Polymer still has some serious problems with internal resistance and has a limited life cycle.
The only advantage to it compared to a standard Li-Ion is that manufacturers can change the shape of it. Outside of that, there is no particularly good reason to switch to polymer if you have a standard Li-Ion battery.
The currently commercialized ones are two technologies (both actually Li-ion-polymer) where "polymer" stands for "polymer electrolyte/separator". Let's call them "polymer electrolyte batteries"
The idea is to use an ionically conducting polymer instead of the traditional combination of a microporous separator and a liquid electrolyte. This promises not only better safety as polymer electrolyte does not burn as easily, but also the possibility to make battery cells very thin as they don't require pressure applied to "sandwich" cathode+anode together. Polymer electrolyte seals both electrodes together like a glue.
Design is: anode (Li or carbon-Li intercalation compound)/conducting polymer electrolyte-separator/cathode (LiCoO2 or LiMn2O4)
Typical reaction:
Both currently commercialized technologies are using PVdF (polymer) gelled with conventional solvent+salt, like EC/DMC/DEC etc. Difference between two technologies is that one (Belcore technology) is using LiMn2O4 as cathode, and other more conventional LiCoO2.
Other more exotic not yet commercially available Li-polymer batteries actually are using polymer cathode, so we could call them "real polymer batteries". For example Moltec is promising to deliver battery with plastic conducting carbon-sulfur cathode. They seem to have problems with self-discharge and other stuff as well as manufacturing cost I guess.
Yet another proposal is to use organic sulfur containing compounds as cathode in combination with electrically conducting polymer (polyaniline for example). This promises high power capability (e.g. low internal resistance) and high discharge capacity, but have problems with cycleability and first of all - production cost.