Why do developing countries recover from crises faster than developed ones?

Because their wages fall faster:

...when it comes to banking crises, the emerging markets, particularly those in Asia, seem to do better in terms of unemployment than do the advanced economies. While there are well-known data issues in comparing unemployment rates across countries, the relatively poor performance in advanced countries suggests the possibility that greater (downward) wage flexibility in emerging markets may help cushion employment during periods of severe economic distress. The gaps in the social safety net in emerging market economies, when compared to industrial ones, presumably also make workers more anxious to avoid becoming unemployed.

One way to understand Keynes macro model is to think about what happens in the labor market when wages are "sticky," and do not fall following a decline in labor demand.  If wages do not adjust, you have involuntary unemployment, and only an increase in labor demand (brought about by an increase in government "stimulus") can bring unemployment back up.

Of course, this "delays" the pain.  Neoclassical economists implicitly want to wait for wages and prices to adjust.  Rogoff's results says that things might be over more quickly if they did.