What is the difference between a battery and a charged capacitor? What is the difference between a battery and a charged capacitor? 
I can see lot of similarities between capacitor and battery. In both these charges are separated and When not connected in a circuit both can have same Potential difference V.
The only difference is that battery runs for longer time but a capacitor discharges almost instantaneously. Why this difference? What is the exact cause for the difference in the discharge times? 
 A: 
Why this difference? What is the exact cause for the difference in the
  discharge times?

The difference is due to the difference in the amount of energy stored.
Consider, for example, a typical alkaline cell.  From this chart, we see that a new alkaline long life AAA cell stores about 5kJ of energy.
Now, consider a large electrolytic capacitor of, say, 1000$\mu F$.  Charged to the same nominal voltage of the cell, 1.5$V$, the energy stored is just over 1mJ.  That's 6 orders of magnitude less.
Put another way, to store the same amount of energy at 1.5$V$ as the AAA battery would require a capacitor of about 4400$F$(!!!).  That's an enormous capacitance.
A: Between the plates of a battery, there is a mechanism (usually a chemical reaction of some kind) that maintains a charge separation across the plates. There is no such mechanism between the plates of a capacitor, so the charge separation cannot be maintained as charge drains off the plates.
A: Some complementary remarks: 
Seen as black boxes, both are simply voltage sources. A battery is designed to supply, as far as possible, a constant voltage whereas a capacitor has no such "dosage capabilities". 
From a applied standpoint, batteries store a lot of charge, but are slow to charge/discharge. Capacitors are the exact opposite: low storage, fast usage. That is why the flash Ina camera uses a battery to charge a capacitor. 
There is also a lot of research in super-capacitors, which try to yield the best of both worlds. 
A: A battery generates a voltage by a chemical reaction. There is a class of chemical reactions called redox reactions that involve the transport of electrons, and you can use the reaction to drive electrons through an external circuit. This is the basis of a battery. The battery will continue to provide power until all the reagents have been used up and the reaction stops. A battery generates a potential difference that is related to the free energy change of the reaction occurring in the battery. Note that there is no charge separation in a battery until you connect it to something and current flows.
A capacitor is completely different. It has a potential only because charge has been stored on it, and when you connect the capacitor to an external circuit a current only flows until all the charge has drained. Unlike a battery, the voltage on a capacitor is variable and is proportional to the amount of charge stored on it.
A: In battery charges are not separated. It is a oxydation/reduction chemical reaction that produces the electrons: ie the current.
In capacitors, the electrons are pre stocked before use during the charge of the capacitor.
