Q:

What's the difference between "Collaborate" and "Cooperate"? 

Both of these words seem to mean much the same thing: working together to achieve some goal.  I can instinctively feel a difference between them, but I can't easily put it into words. 
Can you help me?  Do these words come from different etymologies which might explain the difference? 

A:

Cooperating means working with someone in the sense of enabling: making them more able to do something (typically by providing information or resources they wouldn't otherwise have). 
Collaborating means actually working alongside someone (from Latin laborare: to work) to achieve something. 
The confusion comes from the overloaded meaning of "work with": In the "Work with me, people" sense, it means to go along with my idea - it's a passive condoning or suspension of disbelief rather than an active involvement.  In the "I'm stuck, can you work with me on this problem?"  sense it is a request for active commitment. 
So in terms of helping achieve something, the ordering is something like collaboration, then cooperation, then passive indifference, then active obstruction. 

A:

I think it has to do with ownership of the outcome.  If you collaborate with me on a project, we have shared authorship.  Cooperation could just mean that you've given me help on something I'm working on and that I'm ultimately responsible for. 

A:

If you start with etymologies, you can see that

cooperate
also co-operate, c.1600, from L.L. cooperatus, pp.  of cooperari "to work together with" (see cooperation).  Related: Cooperated; cooperating. 

and

collaborate
1871, back formation from collaborator (1802), from Fr.  collaborateur, from L. collaboratus, pp.  of collaborare "work with," from com- "with" (see com-) + labore "to work" (see labor).  Given a bad sense in World War II.  Related: Collaborated; collaborating. 

share the meaning coming from "to operate" and "to labor", whose meaning in the "co(m)-" sense is almost indistinguishable. 
So, the actual usage is what distinguishes the two words; dictionary entries are almost the same, with the exception from the etymology, that collaborate took a specific meaning during WWII of "cooperate treasonably, as with an enemy occupation force in one's country." 
Also, cooperate can be used for when someone is said to simply "be compliant" - without proactive involvement; where collaborate does imply a bit more active involvement ("cooperate" has slightly wider application, which might be directly related to the fact that the word started to be used almost two centuries before "collaborate").