...between a 3/4 time signature and a 6/8 just by listening to the song? (without seeing the transcription on paper)

same goes for 2/4 and 4/4.
Erm, I think 3/4 and 6/8 are the same, but 2/4 has two beats in a bar instead of four.
You should be able to hear the beats and which beats are accented.
In 3/4 you would hear (in quavers/eighth notes) per bar
1 2 1 2 1 2

In 6/8 they are grouped in 3.

1 2 3 1 2 3.

As for 2/4 and 4/4 it goes (in quarter notes/ crotchets)
4/4
12 1 2

2/4
1 2 | 1 2

The big one is the most accented note, and is at the beginning of the bar, at least that was what i was taught!
Note that "|" is a bar line.

How you count them is personal preference, but the main difference is which beats are accented and how they are grouped together. 6/8 gives a more triplet feel.
Hope that helps.
^ nope, not the same,

3/4 has 3 quarter note beats per bar, 6/8 has 6 eight note beats per bar, 6/4 has 6 quarter note beats per bar

good metallica examples of these 3 would be

3/4 - the interlude "waltz" bit in orion
6/8 - nothing else matter
6/4 - the intro/opening to disposable heroes.
6 eight has a really recognizable feel as well. "ONE two three FOUR five six"
Ah rhythm, perhaps the most misunderstood aspect of music theory.

There are three classes of rhythm (meter). Simple, compound, and hybrid. Simple meter contains pulses of a regular note, like a quarter note, eighth note, half note, sixty-fourth note. Compound meter contains pulses of a dotted note, meaning that each pulse is divided in to three notes. Hybrid meter is a mix of the two.

There are three types of meter. Duple, triple, and quadruple. Duple meter contains two pulses. Triple contians three. And, you guessed it, quadruple contains four pulses. Knowing this, you have a better understanding of rhythm, since each class can be of any type. Simple quadruple is 4/x where x can be any note value. Compound quadruple would be 12/x, because there are 12 beats, but 4 pulses of a dotted note. So in 12/8 one would say there are 12 eighth notes, but the pulses are in dotted quarter notes that contain three eighth notes each. Hybrid triple contains three pulses, two of which are simple, and one of which is compound. So it would be 3 + 2 + 2 (you can rearrange any way) for 7/x.

I hope this introduction to rhythm was helpful, there's still a lot more to know, like writing music in proper rhythm.
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what z4twenny said^

6/8 is like 2 groups of triplets
whereas 3/4 is like 3 groups of eight notes