You do not need to manually save your game in The Amazing Spider-Man. The Amazing Spider-Man uses an autosave system that automatically saves your current progress when you reach save points throughout the game. When a save point is reached, autosave will trigger and the Autosave Icon will appear in the bottom left corner of your screen. Autosave will overwrite older checkpoints with the newest checkpoints.
 
The save points include:
  • Entering or completing a level
  • Reaching a checkpoint
  • Picking up a collectible
  • Buying an upgrade
  • Completing a side quest in Manhattan
The only instance where you can manually save is when you take a picture with the in-game camera. After taking a picture, you will be given the option to save the picture, however taking and saving a picture does not also save your progression.