Simplist Answer - The differences how I would classified them both is -
Non-Specific aka innate immunity has specialized cells like macrophage that go and tell that there are intruders here in the body. They also show the white blood cells and B cells what to look for in pathogen called antigens. Later they can recall and store in memory for next time to fight faster.
Specific immunity aka adaptive immunity works as a specialized Immunity for a particular-pathogen. Such as T-Cells aka white blood cells and cytotoxic T-Cells, they both have differences. Cytotoxic kills off the infected cells before they spread. T-Cells prepares the right formula that was given from macrophages and goes in a Search and Destroy mode.
:) I wrote this in a simplist form with no plagiarism Guaranteed. I am (Confidently) this is correct.