It seems to be a recent trend to point out things you don't like about doctests. There are two articles by Andrew (? - see update below) and one by Ned Batchelder. There's also one by Marius Gedminas.
I take the doctest negativity as a sign of increased popularity of doctesting in the Python world. Doctests are now being seen and read by a larger amount of Python programmers, so there are now more people to talk about the undoubted drawbacks of doctests. (Of course it is also a sign of people disliking aspects of doctests - Marius for one has been exposed to narrative doctests for years)
I like narrative doctests, and have been using them for years now. They often constitute the bulk of the tests of my code. To Andrew at least, that probably means I "abuse" them. Why do I like them?
Doctests are not an ideal testing tool. There are pros and cons. Read the linked articles for some cons (and pros: Andrew points out they are easy to write). Narrative doctests aren't an ideal form of developer documentation either: a well-written, well-maintained dedicated text is better.
The great thing about doctests is that you can write fair tests and fair developer documentation, at the same time. You can use doctests to provide reasonable test coverage suitable for solid, real-world code. Importantly, those same doctests then also provide developer-level documentation that may not always be great, but is still much better than the frequent alternative (nothing).
One advantage of using doctests to describe your API is that you use the API of your code in the doctest before you actually use it for real. As a result, the API of the code you write becomes better as you are forced to think about it early on during the design process. Unit tests of course have the same benefit: improved API design is actually one of the great but rather underacknowledged benefits of unit testing. But doctests encourage you to think about your API design more than plain unit tests, as you're actually writing prose that tries to explain the way your API works to the reader. If it's hard to explain, it may be time to change the design.
Doctests often contain usage examples. Unit tests do too, but doc tests have a narrative around them, including the often all-important setup code. Instead of digging around to see which objects you're supposed to create and what methods you're supposed to call in what order, you have a narrative in the doctest that tells you what to do.
Another advantage of doctests is that they spell out the intent of the tests better than a typical unit test suite does. An individual unit test can of course describe its intent by being well-named and by having comments, but nothing encourages you to do so. Doctests have an actual narrative, so this style of testing actively encourages writing down the intent.
Here are some examples of narrative doctests I've written over the years. I don't find it particularly hard to work with doctest. In all cases below they form the bulk of the tests in the codebase. Is this abuse of the doctest format? Judge for yourself whether you like narrative doctests or not:
Narrative doctests are not an ideal tool; no tool is. You have to actually write a narrative and if you don't, you are left with a testing tool that is in many respects worse than unit tests. I maintain that doctesting is an approach that's good enough to write good, solid software with reasonable developer-level documentation. You can enhance the reuse potential and API design of your library by writing a narrative doctest for it.
Feel free to use unittest and doctest where appropriate, to your taste. But don't be scared off by the recent negativity that seems to surround doctests. Doctests have many benefits. Doctests are a good balance for me personally, and perhaps they will be for you as well.
[update: I don't know why I gave Andrew a last name; I'm not sure where I got that from so I'll take it away again.]