Originally posted by wulf
You'll also find a different set of chords - the Em song will probably feature Em and Bm
The real clue is if it uses B or B7, not Bm!
In answer to the question, the difference between G and Em is the tonal centre. Even if both keys have the same notes, the question is, which one is the tonal centre? What seems to be root note, G or E?
The concept of the tonal centre is a difficult one to explain. It's the key that sounds like 'home', really. It's difficult to put into words, but it's something you learn to recognize.
One clue would be the presence of D#s (in this case). When a piece is in E Minor, according to the key signature, the 7th (D) is natural. However, very often, in pieces in minor keys, the 7th is sharpened, turning chord V into a true dominant chord (e.g. in E minor, sharpening the D turns chord V into B rather than Bm).
Going by the key signature, chord 5 in E minor would be B minor (B D F#). Turning that D into a D# turns chord 5 into B (B D# F#), or B7 (B D# F# A). Dominant chords very often have the 7th added, as it creates a greater 'pull' to the tonic (as the 3rd of chord 5 - D# - resolves to the E, the 7th - A - resolves to the G of chord 1, which is E minor).
So, D#s, or B7 chords, are a good indicator that you're in E minor (or at least, you've modulated to E minor).
I see Wulf has leapt right in there with the old look-at-the-last-chord approach. Lots of people seem to advocate this, but personally, I cringe when I see people recommend it

I also mumble under my breath when I see people say things like "G major will probably end on a G chord and a song in E minor will probably end on an E minor chord"
Music doesn't always do that! For a start, there's quite a lot of music that's in a minor key but ends on the tonic major (e.g. in E minor, ending on an E major chord). Back in the middle ages, you had to do that, or people would start slitting their wrists

There are also songs in major keys, that end on the relative minor (e.g. in G major, ending on E minor). And then there's plenty of music that ends on some other chord, not necessarily even related to the tonality of the piece. And then, there are songs that fade out
Besides which, you shouldn't have to wait until the end to find out what key it was in! You wanna be able to hear the tonal centre much sooner than that. Imagine if you're playing in a jam session! You're not gonna wait till the end to find out what key you should have been in, right?
Forgive me, but looking at the last chord always seems to me to be such a crude way of doing it.
IMO you should learn to recognize the tonal centre without resorting to that.