I need a refresher course. I remember weird shit but forget the basics.
In many cases it is ambiguous, and the people who claim otherwise are trying to make themselves sound important and "in the know".
What's not obvious is what constitutes a quarter and what constitutes an eighth. One man's slow eighth is another man's fast quarter.
True, ultimately the length of a note is going to be relative to the tempo and interpretation of the player, but variation in player tempo is not why we use different meters and there are valid reasons to use one over another that are not just relevant because it makes you sound 'important' to someone who is not 'in the know'
It's supposed to be the way the composer is telling you how they're subdividing the measures. Sure 6/8 might as well be the same amount of space as 3/4. And yeah someones quaver might aswell be another persons crotchet. But when you're writing it out in score, it will often make more sense to use one or the other. (Also in many cases it'll be a lot neater on paper to use a particular choice eg. lots of semi quavers in 4/4 might be a bit messier than a bunch of quavers in 2/2). It can also imply to the player how the piece 'felt' to the composer, and so can help make the piece flow or groove in a similar way to what was originally intended. This is normally related to tempo in correlation with when to emphasise the beat depending on the devising note value.
So while ambiguous at times, 4/4 is not the same as 8/8 (they might be represented this way but time signatures aren't the same as fractions) As others have said, the difference is less in the strict timing and more in the feel or pulse of the music.
Try counting these out loud:
4/4: "1-2-3-4-/1-2-3-4-"
8/8: "12345678/12345678"
2/2: "1---2---/1---2---"
All three of these take up exactly the same amount of time, but to me, the 4/4 time feels steady, while the 8/8 time feels brisk and the 2/2 time feels kind of stately. In all three time signatures, a quarter note will have the same value---one quarter of the measure---but in 4/4 time, it represents the pulse of the music, while in 8/8, it's two pulses and in 2/2 time, it's half a pulse.
A difference in feel between 3/4 and 6/8 time can be even more obvious. In 3/4 time, the measure is broken up into three distinct beats eg. A Waltz. In 6/8 time, by contrast, the measure is often broken up into two beats, each with a triplet feel. Like this:
3/4 time: "ONE and TWO and THREE and/ONE and TWO and THREE and "
6/8 time: "ONE two three FOUR five six/ONE two three FOUR five six "
If the composer writes three quarter notes in 3/4 time, it will feel like three notes on the beat. But if the composer writes three quarter notes in 6/8 time, the second note may feel syncopated relative to the overall pulse of the music.
Usually when playing in 6/8 you will be playing two groups of triplets (in x/4 equivalent) thus making it closer to 2/4 than 3/4. Other than that, you would also notice the difference in terms of strong beats. In 6/8 you would have a strong beat on 1 and a lesser strong beat on 2 (4th eight note) while in 3/4 you would have one strong beat on 1 and two lesser beats on 2 and 3. You would thus notice a difference which would resolve every 3 bars. This can create fairly interesting time changes that won't be hugely abrasive when you you create say; three bars of 5/4 and resolve on a bar of 6/8. DT do this kind of thing all the time except now with MM it's gotten seemingly more specific and deliberate.
If that doesn't make sense then I guess I don't fully understand the differences Haha.. I will say it's certainly not a silly question though, and is something that can easily cause confusion and be quite hard to define, it's much much easier to demonstrate and hear the differences than expressing theoretically.
EDIT: Adjusted some time meters to be more accurate hopefully.
TL;DR: Pointless numbers and shit that don't really mean a whole lot (sounds more complicated on paper than hearing the difference) and you probably don't need to need to know to appreciate the result. I don't have any examples from DT music so consider this analysis kind of irrelevant.