The order of aromaticity of benzene, thiophene, pyrrole, furan, and pyridine I have been taught that the order of aromaticity is: benzene > thiophene > pyrrole > furan. However, I have been unable to deduce any logical explanation for that order. Also, I wish to compare pyridine's aromaticity with these, but my professor does not know about it.

According to me, all the rings are aromatic and satisfy the Huckel's rule of aromaticity with $6\pi$ resonating electrons and a planar ring. So, that point is useless for doing any comparison of aromaticity.

But, we note that $\ce{2p\pi-2p\pi}$ overlap is most favorable in conjugated systems. Hence, thiophene should be the least aromatic (since it has a $\ce{3p\pi-2p\pi}$ overlap instead). 

I agree that benzene should be the most aromatic because both its resonating structures are equivalent and have no charge separation. Using exactly the same reasons, pyridine should be the second best aromatic ring (lesser than benzene because the N atom is an electron withdrawing group and is reducing the electron density in the ring)

That said, every resonating structure (except the uncharged one) for pyrrole and furan will create a formal positive charge on the hetero-atom. Since N is less electronegative than O, it will be slightly more stable than O with that positive charge. Hence, pyrrole will be more aromatic than furan. 

Therefore, according to me, the aromaticity order should be: benzene > pyridine > pyrrole > furan > thiophene.  But it is obviously wrong.

Hence, my question is: what is the order of aromaticity of benzene, thiophene, pyrrole, furan, and pyridine. And why?
The main criteria of aromaticity is resonant or delocalization energy.
It is difference between energy contained in bonds if they were localized like in Kekule's formula for benzene and actual energy.
For benzene if is about 150 kJ/mol.
The more this energy the more stable molecule is.

The approaches for calculating this energy may differ.
As for β-values mentioned in paper by @orthocresol it looks like it is Dewar's resonance energy per one π-electron calculated based on modified Hückel's MO method as mentioned in GSE article. 

There is a table in this article with the same β-values:


In English it is


Benzene                            0.065     Pentalene                     -0.018
Naphthalene                        0.055     Azulene                        0.023
Anthracene                         0.047     Fulvene                       -0.002
Tetracene                          0.042     Pyridine                       0.058
Cyclooctadecanonaene([18]annulene) 0.012     Pyrrole                        0.039
Cyclobutadiene                    -0.268     Furan                          0.007
Azete (Azacyclobutadiene)         -0.160     Thiophene                      0.032