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What is the importance of tempo and time-signatures? Aren't tempo and time-signatures all interchangeable? 4/8 at 60 BPM can be also described as 4/4 at 120 BPM. Can't a 60 BPM tempo be transcribed to 120 BPM just by changing the note durations?
Apart from readability, what other importance is there that a specific tempo or time-signature should be used?
For the most part, the time signature indicates what kind of feel the beat of the piece has. Consider waltzes, usually written in 3/4 – the beat goes ONE two three, ONE two three, ONE two three. Although you could write it as ONE two three, FOUR five six, ONE two three, FOUR five six with a time signature of 6/4, there's no point because the beat still naturally repeats every three beats. The time signature is also chosen by the composer as an aid to the musician to give a hint to which notes might be stressed when reading a piece for the first time.
In response to whether it's possible to rewrite a score with double or half the tempo marking with no change in the music: when a piece is written in cut time (2/2), different conductors will conduct it with either two beats per measure (as written) or four beats per measure (like 4/4 - one quarter note per beat). These two approaches give the music different sorts of feeling – from my experience, a piece conducted in 2/2 usually feels more smooth and connected with much less emphasis on the individual beats.
In short, there's no reason why you can't write a piece with a time signature and tempo different from what might feel natural, just as there's no reason that you can't write a novel without any paragraph breaks, but it makes more sense to do it in a conventional way and makes everyone's life easier.
Simple rule: Changing the bottom number of a time signature, and the tempo correspondingly, has no effect. As you say, 4/8 at 60 bpm is the same as 4/4 at 120 bpm. It's also the same as 4/2 at 240 bpm. Likewise 3/8 at 80 bpm is the same as 3/4 at 160 bpm, and so on.
Changing to top number is a different story, as the other answers allude to. That changes the number of beats in a measure, which can greatly alter the feel. ONE two THREE four is very different from ONE two THREE four or ONE TWO THREE FOUR.
One exception is when the top number is a 6. 6/8 is usually emphasized ONE two three FOUR five six whereas 6/4 is often ONE two THREE four FIVE six, so the bottom number changing results in the rhythm changing. This is probably the most common exception but not the only one — the time signature never determines the rhythm, they just have common rhythms associated with them.
Actually, 4/8 at 60bpm could be written as 4/4 ALSO at 60bpm.
Meters (what the time signature denotes) have two properties: whether they are simple or compound, and whether they are duple, triple, or quadruple (though some argue it is impossible to aurally tell the difference between duple and quadruple). The first property refers to how the beat is DIVIDED - simple beats are divided in 2, and compound beats are divided in 3. The second refers to how many beats are in a measure.
Marches are nearly always in simple duple (usually 120bpm), meaning they have two beats in a measure, and the beats are divided in two. They can be written in 2/4, 2/8, 2/2 (cut time) etc. Semper Fidelis is an example of a march in COMPOUND duple - 6/8. The dotted quarter note gets the beat, and there are two beats per measure, but the beats are divided into three.
In simple time, the possible top numbers of a time signature are 2 for duple, 3 for triple, and 4 for quadruple. The bottom number represents the beat note. In compound time, the possible top numbers are 6 for duple, 9 for triple, and 12 for quadruple. The bottom number represents the DIVISION of the beat (the beat of 6/8 is actually 3 eighth notes - a dotted quarter note).
Now, this explanation is based on the Common Practice Period (1600-1900) standards. Modern composers sometimes break these conventions and even use irregular time signatures. But this is still very useful knowledge.
As for tempo, it is simply how fast the composer wants the perceived beats to go by. As for whether he uses quarter notes, eighth notes, half notes, 128th notes or whatever for the beat or beat division, I think that is simply up to whatever he wants it to look like. :)
