What is the difference between study of מחשבה and השקפה? They both seem to mean perspective on the world...
First, literal translations: השקפה - outlook מחשבה - thought
In terms of fields of study, I think they both refer to the same topic, but with different connotations:
מחשבה carries implications of philosophy, of heavy texts like the Rambam's Guide of the Perplexed, or Rav Hutner's Pachad Yitzchaq, the Leshem's Seifer haKelalim (to include an example from Qabbalah), Rav Tzadoq's writings, the Maharal or the like. It is a deep an involved study, but somewhat cerebral.
השקפה is more experiential knowledge. Less how I think about the world in abstract study and more how I view the world as it happens to me. (As per the translations of the world.) It can be shallower but more poetic, more moving.
At least, that's how the words seem to be used in my neck of the woods.