What is the difference of use between 没空 and 没(有)时间? As far as I can tell, they both have the same meaning: have no time, i.e.:


  对不起，今天我没空帮你
  
  对不起，今天我没（有）时间帮你


But the first one looks like a quasi-idiomatic expression (have no air?); is there a difference in register, perhaps?

Another thing that bugs me: 没 is mostly used for past negation, with the notable exception of 没有. Can I use both of those expressions when referring to the future?
「時間」means time, but「空」means free/leisure/spare time. Correspondingly,「沒（有）時間」means don't have time, but「沒空」means don't have free/leisure/spare time.

While「時間」works in all situations,「空」does not; there are situations where using「空」gives a strange impression or makes little sense, such as


  
  我沒空打工賺錢 - I don't have the free time to go to work to earn money
  我的孩子沒空上學 - My children don't have the free time to go to school
  


In the above, by using「空」, there's a feeling that the task described is trivial or unimportant, which comes across as abnormal if the people around you would consider otherwise.

「沒空」is otherwise more colloquial than「沒時間」, which means that you should use the latter when presenting written work.



Further notes:


「空」here means spare or vacant, and by extension spare time, pronounced in the fourth tone. Air is not the correct interpretation (and is pronounced in the first tone), although it would be related to vacant in meaning.
For「沒」vs.「沒有」, see  What is the difference between 没有+V. and 没+V.?