For some time now I've had a sneaking suspicion that the reason so many developers prefer Context over Drupal's block system boils down to developers taking power away from the client's admin. There may be clumsy aspects to blocks, but client's can understand them. This is well illustrated in this excellent post by Randall Knutson.
Context was the most awesome thing since sliced bread. You could specify conditions and reactions for just about anything. Granular control without programming. We were in heaven. Then we delivered the site to the client. Same old problems started occurring only possibly worse. How do I place a block on a page? Uh, go to contexts and add a context. See, you want to set up your conditions for when it is visible and then add a reaction for blocks... Oh nevermind, I'll do it.
You can read more in How I Learned to Stop Worrying and Love the Block.

Comments
Link fixed.
Link fixed.
Live Better Eclectically!
-- Bryan
Definitely makes sense if you
Definitely makes sense if you have sites where the client wants to learn or mess with the block system. Most of the time it's not about managing content and blocks but creating workflows and process. Context helps the situations where you run into mutual exclusive blocks and block placement. Smaller sites probably don't have this problem but larger ones do. Now, whether sites should be "large" or not is another discussion probably worth having.
Context doesn't shut off the default block system. So both can work and coexist, but then you have two places to manage it from. ;)