As New Historicism emerged (via Foucault, Lacan, Greenblatt, etc.) there was an increase in focus a general diversification of the canon in efforts to acknowledge previous and current minority literature which had been suppressed by the white western patriarchy. Complementing this increase of the canon as far as recognizing previously suppressed minority literature as elite classics; there was also a growing interest in pop culture and so-called ‘low-brow’ literature and art. This had nothing to do with the cultural diversification; it had to do with the change in historicism where all texts (literary and non-literary) were considered ‘literature’ and products of their historical epoch. I mention the canonical inclusion of minority literature and pop culture because they represent alternative literatures (to the elite and the classics) and this was part of the transition from Old to New Historicism: the recognition of alternative perspectives of history.
Old historicism (which I would call pre-New Historicism) is similar to New H. in that the idea is to investigate the historical, social, and cultural world of the author and that these elements are always interconnected with the literature of their time period. Both old and new historicists believe that texts cannot be separated from their historical context.
New Historicism pays more attention to ideology, power and is just more nuanced than its predecessor. With New H., the critic understands that there is no objective history and that ideology plays a role in the work of the critic and the author the critic is analyzing. So, if Foucault is doing an analysis of The Tempest, he would:
1) Recognize cultural, ideological, economic, etc. elements of Elizabethan England and how they influenced Shakespeare; AND, he would note where/when Shakespeare was conscious and unconscious of these elements in his work. He would note that Shakespeare’s work cannot be completely boiled down to Elizabethan culture since history is not that simple. So, Foucault can’t look at Shakespeare as an objective filter of his historical context. He must investigate the historical context, the work and Shakespeare as a subjective interpreter of his own historical period.
2) Foucault would also be conscious of his own subjectivity and historical biases in his investigation of Shakespeare’s subjectivity and historical interpretation.
I think the biggest difference between Old and New Historicism is that New takes a more subjective approach; more conscious of history as interpretation on everyone’s part: author, historicist, historian, etc.