To unlock the Power and Responsibility trophy in Spider-Man you must complete the main story of the game using the new hardest difficulty "Ultimate" added in Update 1.08. You must not change the difficulty at any time during your play through of the main story.
Seeing as this trophy comes with an update that also requires you to finish a New Game + play through you might as well combine your efforts and set your New Game + to Ultimate. This way you get all the advantages of your completed save -- XP level, unlocked skills, suits, mods, gadgets -- to help you adjust quickly to the new difficulty.
Spider-Man isn't a difficult game to complete even on a harder difficulty. The only modifier appears to be enemy damage and damage resistance, yet with plenty of patience Spider-Man can avoid damage entirely even with his basic move set. With all of the unlocks of a New Game + this should be pretty straight forward as long as you take your time.
EDIT: To be clear, you can start a brand new save on Ultimate if you want, you don't have to unlock NG+ first. Also, the game will throw up a warning if you attempt to change the difficulty from Ultimate, so there's no chance of an accidental switch.