A time signature is simply the composer's way of telling you how s/he is subdividing the measure. So in 4/4 time, the composer imagines the measure divided into four beats, while in 8/8 time, the composer imagines the measure divided into eight beats. 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 time: "1-2-3-4-/1-2-3-4-"
8/8 time: "12345678/12345678"
2/2 time: "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 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, as in 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 TRE and/ONE and TWO and TRE and "
6/8 time: "ONE two tre FOR fiv six/ONE two tre FOR fiv 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.
Update: Listen to the first twenty seconds or so of La Pistola y El Corazon by Los Lobos. Is it in 3/4 or 6/8 time? Depending on how you hear it, the music has a much different feel, with the lead guitars putting emphasis on different parts of the melody. Listen a few times and try to hear it both ways!