TITLE: What is the difference between analytic combinatorics and the theory of combinatorial species?
QUESTION [16 upvotes]: Yesterday I asked the question Why should a combinatorialist know category theory?, where Chris Taylor suggested me to have a look at combinatorial species. I had heard the term before but I haven't read more about it that than its entry on Wikipedia. To me it looks essentially like analytic combinatorics dressed in the language of categories. Is this correct? Can we do things with species that are impossible with analytic combinatorics (or vice versa)?

REPLY [5 votes]: I would offer the following difference between the two. 
Species theory offers an extremely formal framework to define a combinatorial class. It is powerful enough to address the question of when does a specification makes sense(*), which analytic combinatorics does not. It is a context in which one can model many analytic phenomena combinatorially, (esp. differential equations) and address meta question like how many combinatorial classes exist. It is a step beyond Polya enumeration, and as such it does not only deal with exponential generating functions, also provides access to ordinary generating functions.
The techniques of analytic combinatorics (say, as in the text by Flajolet and Sedgewick) are far more concrete. They concentrate on a few operators: cycles, sets, etc. and then go much further in the analysis of related generating function. As such, there is more focus on algorithms (and implementations) and methods which work in precise contexts. The singularities tell the story of both counting and parameter distribution, which is remarkable. 
I hope that helps. 
(*) This article translates some of the conditions into remarkably straightforward conditions. http://www.sciencedirect.com/science/article/pii/S0097316512000908
Algorithms for combinatorial structures: Well-founded systems and Newton iterations
Carine Pivoteau, Bruno Salvy, Michèle Soria