First of all, a “Battery” is an array of cells. What we Modern folks call a “Battery” is usually just a single cell. So, Franklin was correct in calling his device a “Battery” - it was a battery of condensers (or capacitors).
a battery means a collection or group, for example an artillery battery is a group of cannons. a capacitor battery is a group of capacitors wired together.
in common use an electrochemical battery is a collection of galvanic electrochemical cells though single galvanic cells are also called a battery.
The electrolysis occurs in the manufacture of the device, not in its operation.
Not unless something has gone horribly wrong. 
It’s called an electrolytic capacitor simply because they contain an electrolytic fluid (i.e. a fluid that conducts electricity). And, as others have pointed out, they use electrolysis to make them. Electrolytic capacitors are really quite clever. All capacitors have the same general form:
conductor | dielectric (insulator) | conductor
A traditional capacitor would do this:
metal plate | dielectric | metal plate
An electrolytic capacitor looks like this:
aluminum | aluminum oxide | electrolytic fluid
Physically, it looks like only two layers. The aluminum conductor simply gives way to a thin outer coating of aluminum oxide (i.e. aluminum “rust”) that acts as a dielectric.The electrolytic fluid can store and release charge much like a metal plate. So you still have all the components of a traditional capacitor, they just take a somewhat unusual physical form.
Haha, I was just about to mention that in the same context. It’s been a (thankfully) long time since I opened up a computer case and saw crusty orange goo.
We misuse the term “battery” today. A AA battery isn’t a battery. It’s a cell. AA, AAA, C, and D are all cells. A 9 volt is a battery. Open it up and you’ll find six cells inside of it. Good luck trying to convince your average person on the street that a C cell isn’t a battery though.
A capacitor and a battery both store energy, but they are only superficially similar. A battery tries to be a constant voltage source, but in reality ends up being more like a constant voltage source in series with a resistor.
A capacitor is nowhere near a constant voltage source. The voltage and current relationship in a capacitor is a simple integral/differential type relationship. There are plenty of dopers here who understand calculus and get this right away, but for the benefit of those who only know algebra, basically if you keep feeding current into a capacitor the voltage is going to keep increasing. Contrast this to a battery where the voltage is going to stay mostly constant. Likewise, when the battery discharges, it is going to keep a relatively constant voltage until it gets almost completely discharged, at which point the voltage suddenly drops like a rock. A capacitor discharges on an exponential curve.
Typical capacitor discharge curve:
Typical battery discharge curve:
As you can see, they are dramatically different.
You can use a capacitor to store energy and power something, just like (well, more like similar to) a battery. To do this, you have to stay up on the high end of the capacitor discharge curve so that your voltage doesn’t drop too low. VCRs in the old days had a battery to backup the clock (or else they just flashed 12:00 if the power blinked). More modern VCRs and now DVD players and the like typically use a capacitor backup. There are two reasons for this change. The first is a fairly dramatic improvement in capacitor technology in the past twenty years, which allows a fairly small capacitor to store a lot more energy than before. The second is the use of lower and lower power circuits as technology in general has improved. These two factors combined have allowed electronic device manufacturers to use cheaper capacitors where batteries used to be required.
Note however that even though capacitors are sometimes used like batteries to store and release energy, they are doing it in different ways. Batteries store the energy chemically. Capacitors store the energy in an electric field. It’s completely different ways of accomplishing the storage and release of energy.
Capacitors and batteries can both be used as filters. Although they end up accomplishing the same goal, just like above, they are using two different methods to do it. Car batteries have been used for years to smooth out the output of a typical car alternator. Capacitors in power supplies perform exactly the same function. However, even though batteries and capacitors are doing the same function, they are doing it in different ways.
Power factor correction highlights how batteries and capacitors are different, though. Inductors and capacitors are both energy storage devices. Inductors store energy in magnetic fields, and capacitors store energy in electric fields. Because of this difference, if you are on an AC circuit, capacitors are charging while inductors are discharging and vice versa. Your typical home tends to run a bit inductive, due mostly to motors (vacuum cleaner, refrigerator compressor motor, etc). The power company balances this out with large capacitor banks, since if they can get the capacitances and inductances to exactly balance then the power transmission becomes much more efficient. You can’t stick a battery in place of a capacitor in a power factor correction circuit. Batteries just don’t function the same way.
Similarly, in an AC circuit you can use a capacitor in series to cut down the voltage going into a device without generating heat like a series resistor would do. A battery wouldn’t work for this. You could do a similar thing with a battery on a DC circuit, where a capacitor would simply stop all current from flowing.
So there are clearly some applications where capacitors and batteries behave completely differently.
You can use them similarly in certain types of applications, but batteries and capacitors are not at all the same thing.
Ha! - simultaneous post with the above. 
Between them Joe Frikin Friday and Stathol have it complete.
Just to reitterate. A capacitor (also once called a condenser) is a fundamental electrical device. It stores energy with an electrostatic field. As Joe said, it has a defined charge to voltage law. The more charge you put into a capactitor the more voltage appears across its terminals. This is so until you reach the physical limits of the construction, when typically the dielectric fails.
What we call a battery (and has been mentioned is actually an electrochemical cell - but a group of them is a battery - in the same way that a group of guns firing together is a battery.) Cells store energy by a chemical process. The reduction potential of the chemicals chosen in the cell determine the voltage the cell provides, and it is fixed. Second order efects in the construction of the cell result is some small variation in voltage, but through the cell’s working range the terminal voltage is essentially the same.
Although Electrolytic capacitors contain water and a chemical cocktail, no chemical reaction occurs in operation. The fluid is simply one plate of the capacitor. The fluid is chosen so that it conducts well, and doesn’t degrade with heat too much. The trick with the electrolytic capacitor is to get a huge surface area, which is done by etching the surface of the aluminium foil. Over time the manufacturers control of the etching process has allowed them to produce more and more fine grained and deeper etching, and has led to substantial gains in effective surface area, and thus smaller capacitors for the same capacity. This etching is also why the opposite plate needs to be a fluid - it is required to flow into the etched surface to make intimate contact with the dielectric (aluminium oxide) created on the surface.
Supercapacitors take the idea of the electolytic capacitor and use a 3D structure to make one electrode. This yields phenominal capacity.
You can’t interchange batteries and capacitors in a circuit. It simply doesn’t work. Capacitors have other interesting properties - in particular they are used in alternating current circuits where the phase angle between current and voltage is important.
It is worth noting that one of the other fundamental circuit elements - the inductor - also stores energy. It uses a magnetic field to do so. Nobody suggests that it looks like a battery. (A superconducting inductor makes a perfectly good storage device, if a little unweildy and hard to manage.)
I remember when those big value, small size capacitors became readily available. It was pretty weird see a capacitor labeled in farads when most people working in electronics had never seen anything bigger than maybe a few 1000 microfarads.
I think they were (still ?) used as the backup “battery” for the clock in VCRs etc.
This is not the reason it works. It works because the various items supplied from the fuse box in the car normally share one of 1-3 runs between the fuse box and the alternator or battery. This couples noise into the radio power leads. Dedicating a run to the battery decouples the radio from these glitches. Among the glitches may be alternator noise related to the battery charging current, which such a dedicated run also eliminates.
Saying a battery and a cap are the same becoust they both can store energy would be lik saying a house and a high rise building are the same because people go into both.
Saying a cell is not a battery is like saying that 1 is not a number.
Ultracapacitors powerful enough to run cars are an active area of research if Technology Review is to be believed.
I know I’m being totally pedantic here, but:
There is a distinct use of plurals
Modern usage has corrupted the word “battery” to refer to what is actually just a single cell, but that doesn’t really make the usage correct. 1 is certainly a number; however technically speaking, you cannot have a “battery” of one cell any more than you can have a “flock” of one bird.
A car battery is indeed a battery because it contains multiple lead-acid cells. But a “size AA battery” is actually a “size AA cell”. Energize, et al. are bad about perpetuating this confusion, but if you ever shop around for non-consumer NiMH cells, or similar, you’ll find that the vendors always refer to them as cells and only use the term “battery” to refer to several cells working in conjunction.
It depends how you define “basically”.
If I were showing them to an alien I could describe each as an energy storing device for later use. Sure, one stores a lot and releases it slowly, the other stores less but releases it quickly but I don’t think our green friend would have trouble with the ‘basically the same’ concept. Or so it seems to a lay person like myself.
Much like one could argue that all religions are basically the same. While it could be equally countered, “completely untrue”. Depending on context, both are true.
Once again, untrue.
“Basically” means that at the most fundamental level, these two things are more-or-less the same. But, that’s not the case; they are only superficially the same once you look at their functionality.
You wouldn’t say that an automobile and an airplane are “basically the same,” would you? After all, they are both machines, and they both move people.
Or is a Steak and a cupcake “basically the same?” After all, they both provide nutrition.
The OP asked if a Battery is a capacitor.
It isn’t.
No matter how many people want to try to make it into one.