I would like to know why some capacitors have the same value (capacitance) but their sizes are different? What is different between those capacitors?
They will probably have different dielectric, meaning different working temperature and tolerance. See table here: http://en.wikipedia.org/wiki/Ceramic_capacitor#Class_2_ceramic_capacitors
Also, bigger capacitors will usually have higher voltage rating, they cool down better.
It also might be age (caps get smaller with years) or manufacturing capabilities. For example of the latter: if you were to buy strictly "Made in Russia" parts, you'd have to tolerate with much larger packages for the same thing, say, Murata makes.
Sometimes (or even usually) there is no real difference, so you can choose depending on the size itself: if you solder by hand, bigger size can be an advantage.
I also remember reading one interesting app-note, focusing on Capacitance as a function of DC Voltage. Generally, physically smaller caps "degrade" more. You can find it here: http://www.maximintegrated.com/en/app-notes/index.mvp/id/5527
For a given (fixed) set of constraints:
- Manufacturing technology,
- Dielectric type,
- Target application, i.e.: decoupling, general purpose, high-frequency or power line filtering,
- Mounting style, i.e.: SMD, through-hole or chassis,
- Capacitance value,
The only feature that requires increasing the size of a capacitor is its voltage rating.
Reasoning the other way around,
You can trade off a smaller voltage rating of the capacitors in your design for a smaller package size (assuming the set of constraints above).
The capacitance C between two plates of area A, separated by distance D, having a dielectric with relative permittivity Er is...
C = Er * E0 * A / d
Where E0 is the permittivity of free space.
If the plates each have thickness t then the volume V of such a capacitor is ...
V = A * (d + 2 * t)
Ceramic capacitors are made of many very thin layers of alternating metal and dielectric stacked together. If a ceramic capacitor has N plates then it has a total volume V of...
V = A * N * t + A * (N-1) * d
Each dielectric will have a different Er value. For example, X7R may have Er of 3300 but may drift 15% over temperature. Whereas NP0 may have Er = 120 but may drift only 30ppm/C.
Therefore the total volume of the capacitor depends on what dielectric is used and how thick we make the electrode plates.
If you want the capacitor to handle more current or have lower ESR then the thickness of the metal layers needs to be increased.
The breakdown voltage of a dielectric layer is proportional to the thickness of the layer. Therefore making thicker layers may create capacitors with larger voltage ratings.
The choice of dielectric involves a trade between how much capacitance we need in a given area, how much capacitance drift we can tolerate vs. temperature, and the required voltage rating.