This question sounds hard, but it's actually very easy to answer :-)
Harmony is vertical based and counterpoint is horizontal based.
In the old days, when mr. Bach was doing his thing, harmony as we know it didn't really exsists in the same way. Ofcourse, multiple-sounding-notes are creating a harmony. But it wasn't a harmony like we hear now in pop and jazz (etc...)
So: if you have a jazz tune in C and the melody goes: E-F-G and the bass player plays C-D-E then there is a big change the piano/guitar player plays: Cmaj7, Dmin7, Emin7.
This is vertical.
If Bach wrote: E-F-G and wrote in the bass C-D-E he didn't mean Cmaj7, Dmin7, Emin7 he just meant them to go together.
This is horizontal
Then there is something else, counterpoint is a set of rules most of them influenced by church. Google for consonant, dissonant etc. Nowadays you can do whatever you want (thanks to enlightenment! Jeej!) but back in the days, you couldn't.
So, why studie counterpoint and harmony?
Well, harmony is obvious; music of today works on 'harmony'
Counterpoint is nice to studie so you can learn to work in a framework. Maybe you're even able to create your own! Like the 12 tone system Schoenberg created (http://en.wikipedia.org/wiki/Twelve-tone_technique)
* - if you do this, please mention it at music.stackexchange ;-)
Then I would like to end with two youtube clips.
1) It's a fuge (a fuge is a way of composing, also rules. But used in the language of counterpoint)
2) Enjoy the lines played by Glenn Gould, you can hear them move together!