I'd guess that that blur is a raster image. Are you certain this is vector?
In order to do something similar in Illustrator you'd need to use Gradient Meshes. It's far too complicated to try and explain on a message board, but if you search Google for "Illustrator Gradient Mesh Tutorial" you'll find many of them.
Black and white gradient, with an envelope distort then used as an opacity mask over a white block could do this I think.