Last Positivist recently had a very good post, Learning from Four Analytic Philosophy Wins; except for the 'Statistical Method' examples, which I think are much less clearly wins than they would usually be regarded, I am pretty much in agreement with it. But I think there is another key aspect of analytic philosophy, which provides, overwhelmingly, most of the good in it: it is very good for tinkering with arguments. Everything in analytic philosophy facilitates argument-tinkering. Arguments are broken up, premises identified, logical connections identified, each part is often discussed at length, different variations are considered.
The value of just tinkering with arguments should not be underestimated. You can learn a great deal about reasoning that way, you can refine and improve particular arguments to a high degree, you can sometimes turn probable inferences into proofs. It can help in classification and in seeing how arguments relate to other arguments, particularly since both can depend on fiddly bits of arguments that need to be identified, compared, and contrasted. And so forth. It all requires tinkering in the right way -- but if you look to that, analytic philosophy makes the rest easy.
Argument-tinkering is not the only way to approach philosophical questions; most approaches to philosophy do not put a great deal of emphasis on it. Indeed, there are only very few that do. But they put analytic philosophy in very good company; the primary other approaches that are built with the principle of tinkering are the family of approaches we call 'scholasticism' and the Nyaya-Vaisheshika school of Indian philosophy (with some overflow into other schools interacting with it), and these are very good company indeed. The comparison shows one of the strengths of the principle of tinkering: a lot of power with a lot of flexibility, if used well. And analytic philosophy arguably does tinkering better than either.
Of course, tinkering has its weaknesses, too. Arguments are not isolated machines. Your response to a syllogism in philosophy of science may commit you to a position in political philosophy; your views on metaphysics may imply things about aesthetics. The connections are not always obvious, but that just means that you have to actually put some work into uncovering them. Analytic philosophy has always struggled with this; in their objections and responses, analytic philosophers are always making claims that they never adequately think through, whose ramifications in other fields they never properly follow up on. Tinkering with individual arguments is not what you need for such thinking. But this is not an inevitable problem, and there are people who avoid it; you just have to recognize that tinkering, valuable as it is, only gets you so far.
