Opinionated software means that there is basically one way (the right way™) to do things and trying to do it differently will be difficult and frustrating. On the other hand, doing things the right way™ can make it very easy to develop with the software as the number of decisions that you have to make is reduced and the ability of the software designers to concentrate on making the software work is increased. Opinionated software can be great to use, if done well, if your problem maps onto the solution nicely. It can be a real pain to solve those parts of your problem that don't map onto the tools provided. An example here would be Ruby on Rails.
Non-opinionated software, on the other hand, leaves lots of flexibility to the user (developer). It doesn't proscribe one method of solving a problem, but provides flexible tools that can be used to solve the problem in many ways. The downside of this can be that because the tools are so flexible, it may be relatively hard to develop any solution. Much more of the solution may have to be hand-coded by the user (developer) because the framework doesn't provide enough help. You also have to think much more about how to provide a solution and mediocre developers may end up with poorer solutions than if they had bought into some opinionated software. PERL is probably the classic example of non-opinionated software.
My ideal is a non-opinionated framework, but one with strong conventions. I would put ASP.NET MVC in this category. In reality all software is opinionated to some extent (though perhaps not PERL). MVC has strong conventions in its choice of model but offers many different ways to solve problems within those conventions. Some of those ways may even break the model. Used correctly, however, in accordance with the conventions developing in such a framework can be a real joy.
