To assess something would be when you want to get an overall picture or idea of a condition of something, or of a situation as a whole, whereas to examine something typically implies that you're looking at an object for something(s) specific.
-The food inspector stopped by to assess the condition of the restaurant, and to make sure there were no food safety violations being committed.
-The appraiser was hired by the buyer to assess the total value of the house before purchasing.
-After the last hurricane, the damages were assessed to be in the billions of dollars
For an examination or to examine, some examples would be:
-The body was examined for signs of foul play
-The diamond must be examined for flaws and imperfections before it can be given a price
-Please examine this paragraph for me and see if I made any mistakes
Hope this helps