You're playing an instrument with no bass. If you were playing them on guitar, or piano, you would hear the difference more readily.
The Em and G have nearly the same notes, as was said, and in fact, the Em7 and G6 have the same notes. But, Em is usually played with E in the bass and G near the top, whereas G6 is played with G in the bass and the E near the top. But if you have no bass string, you miss out on this difference.
For pianists, the note in the bass is the note the chord is named after, so EBGD is Em7 and GDBE is G6.
If you have a bass player, let him alternate between E and G while you strum the Em7/G6 on uke (0202).