Applying for both assistant and associate professor position at the same university I'm an established, normal, non-stellar postdoc in computer science, several years after my PhD defense, and am going now for higher-rank academic positions. A particular university in the US offers both assistant and associate professor positions in my subfield. These are two different positions, but the announcements are copy-and-pasted with small adaptations. I would like to apply for both, since I wish to maximize my chances at a small additional cost (about an hour of adapting the cover letter and submitting).


Is it normal to apply for two positions simultaneously? 
Is there anything particular to be paid attention to in the application documents (cover letter, research statement, teaching statement, reference list, publication list)?
What are the typical pitfalls of job seekers in such a case?
First of all, are you sure you have a clear idea on the distinction between an assistant professor and an associate professor at a US institution, and in particular at this US institution?  

The majority of associate professorships come with tenure, so that's the first thing to find out.  To go from a postdoc position to a tenured position on arrival is possible but really rare -- this is something that happens to someone because they're a superstar, not because they went the extra mile in applying for more positions.  To apply for both a tenured position and an untenured position sounds a bit weird to me: are you qualified for and desiring of tenure on arrival or not?  If the associate professor position does not come with tenure, it is not completely clear what the advantage to you in taking it is.  At all US institutions I know, the responsibilities and rights of all tenure track faculty are essentially identical (exceptions include certain faculty votes).  Moreover, in my department the average salary for assistant professors is sometimes higher than that for associate professors, because the annual raises do not keep up with market value.  At most institutions I know of you get a fixed, automatic raise upon promotion from assistant to associate and from associate to full, so arriving as an assistant professor with a short tenure clock might be more lucrative than arriving as an associate professor.  

I would recommend against doing this "just to economize."  To address your specific questions:


  Is it normal to apply for two positions simultaneously? 


It is not so abnormal, anyway.  I have often seen people apply for both postdoc and tenure track assistant professor positions.  Actually though this makes more sense to me, because these are both natural continuations of a PhD or postdoc position.  But I can't think of anyone who has applied for two positions and been awarded the higher position.  (I applied to my present institution for a postdoc the year before I applied as a tenure track assistant professor.  I didn't get the postdoc position, but I did get the tenure track one the next year!)

I think most people's reaction to that will just be to consider you for whichever of the two positions seems most appropriate to them.  Conversely, if multiple positions are available and you apply to one position, if those who read your application think you are better suited for another position, they may invite you to apply for it or even just carry over your application.  


  Is there anything particular to be paid attention to in the application documents (cover letter, research statement, teaching statement, reference list, publication list)?


Presumably the documents would all be identical except for the cover letter.  If you actually have a good rationale to apply for both, the cover letter is a good place to explain this.  


  What are the typical pitfalls of job seekers in such a case?


The main risk is along the lines I outlined above: it makes you look a bit naive and unfamiliar with the nuts and bolts of academic jobs.  But most young academics are this way to some extent, so I don't think it's a terrible risk.  On the other hand, I don't see much reward either.

All of this is a great thing to talk to your advisors / mentors about, by the way.  If you're really a good fit to go straight to an associate professor job, they'll know and be happy to tell you.