I agree: Bragg's book is certainly excellent (I've not seen the series), but he is a presenter, albeit a more scholarly one than many of his peers, and it shows in the level of detail in his book compared to Crystal's, who is an academic. For example, in one of the Old English chapters of "The Stories of English", I seem to recall Crystal analyses several versions of a manuscript line by line, identifying potential scribal errors and dialectal variation. Great stuff!
So I'd say that if you want an easy bedtime read, Bragg's book is for you (or perhaps Bill Bryson's Mother Tongue: The Story of the English Language, which I've not read); if you want a slightly less accessible but more thorough discourse on the development of English, Crystal is your man. He's written other books on the English language as well, as I'm sure have other academics.
David