I'd mark this a duplicate, except (a) I'm not 100% sure I understand what you mean by "separating", and (b) what "the same thing as clicking on the ZIP icon" is will very much depend on whether that button performs a dynamic git archive on demand, or whether the archives are pre-generated (in which case they might not match the current HEAD).
(I would imagine that they are dynamic, however; or at least that github invalidates any cached copy if it is no longer current).
What does "checkout is not working" mean?
Anyhow, if you wish to clone a specific branch without downloading the other branches, that's been asked and answered: https://stackoverflow.com/a/7349740/324105
or if you're using git 1.7.10 or later: https://stackoverflow.com/a/14930421/324105