I'm confused as to what a cadence does. The only thing I understand is that like English it's a period in a sentence. Is that its primary purpose to separate pieces of thought? And how important is it? Is it possible to play music without cadences?
Just see how many different sentences you used in these three lines you posted! 5 Different ones, which where separated by punctuation marks. Think of music as a language (which it basically is). It would be weird for a person to speak and not use any punctuation marks, to differentiate the sentences; music works the same way, with many kinds of cadences (not exactly the same, but in the same sense).
A cadence is just like that. You play (say) a musical sentence and then you take a breath and continue talking. It gives a sense that the music has stopped for a while or has finished all together; it separates the things the music wants to say.
Of course, you can talk without differentiating the sentences, if that is what you are going for. It's a technique that is being used in literature, speaking and musical composition/improvisation.
Furthermore, in a purely musical context, the cadence also indicates the tonality of the musical phrase.
A cadence does at least two things. It makes a statement about what key you are in at the moment. It also indicates a completion of a thought and therefore the beginning of another, similar as you point out to sentences or clauses in writing.
But cadences can also be avoided. In his opera Tristan and Isolde there is a scene where the composer Richard Wagner avoided using a cadence for about thirty minutes. It's called the "Lovedeath" scene, because in it the two protagonists are dying and taking their long sweet time about it. Wagner keeps up the suspense of 'when are they finally going to die' by extending the music through many different keys and avoiding anything that sounds 'final' until the very end, when they do die. He achieves this by veering off in a different harmonic direction every time a phrase sounds as though it is about to experience a cadence and therefore come to a close. It's a remarkable achievement and laid the groundwork for atonalism and a lot of other developments in music. This scene alone has influenced many other composers, including a lot of people currently writing musical scores for movies. They use the same technique to develop suspenseful sounding music for scenes in which that is appropriate.
In a backwards sort of way, then, this enormous example shows why cadences are important. By avoiding them for such an extended length of time, Wagner say, in effect: "Cadence are important to the listener because we have come to expect them to show the grammatical construction of our language. By avoiding them in this scene, I show how a thought can be extended indefinitely, or for as long as the listener can stand it." Some listeners would say "for longer than I can stand," but that's beside the point. Think of a whole chapter in a novel which consists of one long sentence. It would be difficult to read but would get the point across that the scene in the novel was rather 'endless' in a sense. That's Wagner's point, and in making it he tells us something about our abilities as listeners.
Cadences articulate musical phrases.
If you follow this idea a few steps, you will see that
V - I is not a cadence... unless it ends a phrase.
But, I won't say any more, and instead point you to this great article where I read it.