Do Ampère's and the the Biot-Savart law give different answers for the magnetic field of a circular loop? I have been trying to calculate the magnetic field due to a circular loop using Ampère's circuital law. However, I am unable to do the same. 
Whenever I apply the law, while integrating length element $\mathrm{d}l$, I get $2\pi r$ [the circumference of the loop]. However, when we apply Biot - Savart Law, the $\pi$ from the circumference gets cancelled with the $\pi$ from the constant of proportionality. Please do point out the error in my derivation. 
I would also like to know if the usage of Biot-Savart law is any different from Ampere's Law when considering the restrictions or required conditions.
 A: Ampere's law is not useful in this case. It says that the line integral of the B field around a closed path is equal to $\mu_0$ times the current passing through the closed path (for steady currents).
To use the law you want the LHS to be simple to evaluate. Usually the B field is constant in magnitude around the path and either parallel or perpendicular to the path. In this case you cannot arrange this. As you can easily show with the Biot-Savart law, the B field varies with distance from the centre of the current loop, so it is difficult to define a simple line integral path that encloses the current.
A: It is not at all obvious how Ampère's law would be used to to calculate the field at the centre of a circular loop of wire. Ampère's law tells that if we draw any closed curve in space then the magnetic field integrated along that curve is proportional to the current flowing through the space enclosed by the curve. Typically this is useful where we can choose a curve such that the magnetic field is constant along its length, because then the result of the integration is just $\ell B$, where $\ell$ is the length of the curve. I can't see any way to use this technique to calculate the field due to a circular loop of wire.
Your Biot-Savart result shouldn't contain $\pi$. You should get:
$$ B = \frac{\mu_0 I}{2r}  $$
