The ICM website records all the addresses given at the Fields Medal conferences, and they always begin with a short paper by a distinguished mathematician describing to a "broad" audience why that medallist has deserved the award. This is often very interesting, containing recent historical remarks. 

You can read Katz' description of Deligne's work in 1978, for instance: I find it delightful. Funny enough, the very same conference saw a contribution by André Weil
http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0227.0236.ocr.pdf
entitled "History of Mathematics: Why an How" which addresses many of your questions – and answers them, I think. For example, he discusses "why" is a mathematician interested in the history of "which" mathematics, and "how" this should be pursued. And the very last paragraph begins: _"What, then, separates the historian from the mathematician when both are studying the work of the past?"_ Besides all, extremely enjoyable!