Polar moment of inertia vs moment of inertia can differentiate as the polar moment of inertia shows the resistance offered by an object to deformation by torsional load and the moment of inertia shows the resistance offered by a crosssection of the object to the deformation (caused by bending load).
Contents:
Polar moment of inertia:
The Polar moment of inertia is associated with the crosssection of an object that shows the resistance offered by an object to deformation caused by torsional loading.
It is one of the types of the second moment of area that is generally denoted by the symbol ‘J’.
The SI unit of the polar moment of inertia is m⁴ and the FPS unit is Ft⁴.
From the above diagram, the Polar moment of inertia about the zaxis is given by,
`J_{Z}` = `\int` r².dA
 We also have a detailed article on the Polar moment of inertia where you can read more about this with solved numerical.
Moment of inertia:
The moment of inertia shows the distribution of crosssectional area from a specific reference axis. It is also known as the planar moment of inertia and it is one of the types of the second moment of area.
The moment of inertia represents the amount of resistance offered by the crosssection to the deflection due to the bending loads.
It is generally denoted by the symbol ‘ 𝙸 ‘ and the reference axis for finding a moment of inertia always lies in a planar surface.
For the crosssection shown in the above figure, the moment of inertia about the xaxis is given by,
`I_{\text{xx}}` = `\int` y².dA
And the moment of inertia of the above crosssection about the yaxis is given by,
`I_{yy}` = `\int` x².dA
Difference between moment of inertia and polar moment of inertia:
Sr. No.  Polar moment of inertia  Moment of inertia 

1]  The polar moment of inertia is the resistance offered by the crosssection to torsional load.  The moment of inertia is the resistance offered by the surface to the bending deformation. 
2]  The polar moment of inertia is taken about an axis perpendicular to the plane of crosssection passing through the centroid.  The moment of inertia is calculated for the axis lying into the plane of crosssection. 
3]  The polar moment of inertia is used to analyze torsional loading in shafts.  The moment of inertia is used for analyzing beams or other flexural members subjected to the bending load. 
4]  The polar moment of inertia is denoted by the symbol ‘J’.  The moment of inertia is generally denoted by the symbol ‘𝙸‘. 
5]  The torsional stiffness (K) depends on the polar moment of inertia.  The bending stiffness of the member depends on the moment of inertia. 
6]  For the same crosssection, the polar moment of inertia is greater than a moment of inertia.  For the same crosssection, the moment of inertia at any axis is smaller than a polar moment of inertia. 
FAQ’s

What is the difference between moment of inertia and polar moment of inertia?
The moment of inertia states the resistance offered by the crosssection for the bending load and the polar moment of inertia states the resistance offered by the crosssection of the object to the torsional load.

How moment of inertia is related to polar moment of inertia?
The relation between the moment of inertia and the polar moment of inertia can be given by using the perpendicular axis theorem.
J𝗈 = 𝙸𝗑 + 𝙸𝗒
Where,
J𝗈 = Polar moment of inertia
𝙸𝗑 and 𝙸𝗒 = Moment of inertia about mutually perpendicular axes (x and y axis) which are lying in the plane of crosssection. 
What is the relationship between moment of inertia I and polar moment of inertia J for a circular section?
For circular crosssection, the relationship between the moment of inertia and polar moment of inertia is given by,
Polar moment of inertia (J) = 2 x (Moment of inertia about centroid)
As 𝙸 = (π/64)D⁴
∴ J = (π/32)D⁴ = 2 x (π/64)D⁴ = 2 𝙸