Is superconductor just a perfect conductor or anything more than that? If I had a hypothetical perfect conductor having infinite conductivity and I cool it below a certain temperature, will it be a superconductor? If not, then how can we distinguish between the two using the experimental and theoretical methods?
I only know the following thing from Wikipedia:

Distinction between a perfect conductor and a superconductor
Superconductors, in addition to having no electrical resistance, exhibit quantum effects such as the Meissner effect and quantization of magnetic flux.
In perfect conductors, the interior magnetic field must remain fixed but can have a zero or nonzero value. In real superconductors, all magnetic flux is expelled during the phase transition to superconductivity (the Meissner effect), and the magnetic field is always zero within the bulk of the superconductor.

Please clarify for me about this doubt and let me know if there is something additional to it.
 A: I can see two big differences between a hypothetical perfect conductor and a superconductor:
1. A "perfect" conductor will keep its magnetic flux condition. A superconductor will always expel the magnetic flux (Meissner effect).
What does this mean?
Let's say that you have a regular conductor in an external magnetic field, which, of course, is also inside the material. Now, somehow, out of the clear blue sky, this conductor becomes "perfect". The magnetic field will be kept inside the "perfect"conductor as when it was "regular". Now, turn off the external magnetic field. The field inside the perfect conductor stays the same.
Let's do the opposite. No applied magnetic field when the material was regular. It becomes perfect. Now, turn on the field. The "perfect" metal adapts itself to have no field inside it.
For a perfect metal, in short, whichever magnetic field was inside it when it was "regular" will be kept in the "perfect" state.
A superconductor behaves differently. Whatever its initial state, when the material goes superconductor it expels the magnetic field. The magnetic field inside a superconductor will always be zero (well in Type II superconductors there is a mixed state, but that is another story).
To account for this difference, the London brothers developed an addition to Maxwell equations.
2. Second difference: The superconducting gap
Roughly, superconductivity comes to life because electrons form pairs (Cooper pairs), which have a lower energy when compared to two isolated particles. This pairs can go into a single quantum state that represents superconductivity. The energy gain per electron in the Cooper pair is called the "superconducting gap". If you give that energy back to the superconductor, the pairs will be broken and the superconductor comes back to be a normal metal. You destroy superconductivity.
You can give this energy in several forms. The first, obviously, is warming the material. The critical temperature of a superconductor is the temperature that is equivalent to the superconducting gap. You can also apply a high magnetic field so that the magnetic energy goes above a threshold (the critical field) that kills superconductivity. Or make a very large current go through the material. Above some value (the critical current) superconductivity will cease to be. Even light, intense enough, will destroy superconductivity by giving the photon energy to the Cooper pair and breaking it apart.
A perfect metal would not have such a "superconducting gap".
A: There is of course no such thing as a perfect conductor.  But if there were, the answer would be "it depends."  Various normal conductors become superconductors below some temperature; many other normal conductors do not.  Since we can only theorize about the extraordinary material which makes up a perfect conductor, we can hardly make a claim as to whether Cooper Pairs will suddenly show up.
A: Indeed, zero resistivity(infinite conductivity) is not taken as the true definition of superconductivity. The fundamental proof of a superconductor is the demonstration of Meissner effect. Thus, a superconductor is not just a perfect conductor having infinite conductivity but also a perfect diamagnet that exhibits Meissner effect. For your question a normal metal will not become a superconductor even if you cool it down to a low temperature. To distinguish them experimentally, you can search for "quantum magnetic levitation" which shows that a magnet can float above a superconductor(at low temperature) and this phenomenon is a demonstration of Meissner effect.
Theoretically, phase transition occurs when a normal metal becomes a superconductor at the critical temperature $T_c$. Just like the phase transition between water and ice, the properties of the two phases are very different. But if there is no phase transition, we would not obtain a superconductor no matter how cold it is. For the details of the microscopic theory you may search "BCS theory".
A: The main difference between superconductor (SC) and perfect conductor:
In a SC the kinetic energy of every charge carrier is quantized, so a supercurrent, once established, can be dissipated only by external energy excitations stronger than the kinetic energy quantum. In a perfect conductor the kinetic energy of every charge carrier can increase/decrease smoothly, by arbitrarily small values, so the current dissipates due to vanishingly weak energy fluctuations (thermal, electromagnetic etc.) The experimental method to distinguish SC and perfect conductor is trivial: SC can keep a supercurrent forever, a perfect conductor cannot (despite a vanishingly small resistivity). Note, the Meissner effect is a direct consequence from the persistent supercurrents in a SC.
