It does not matter how much you rub and scrub at these marks, when they are there they seem to be determined to be a bathroom feature forever, even when you use proprietary cleaners that claim that they will rid your shower screen of stains.
Do not despair, though - there is an effective way round this problem that will get rid of the streaks and the water spots yet will not cause any damage to your glass shower door.
Spray WD-40 onto the glass and gently rub it round the glass so that all of the stains are covered in it. Leave it for a minute but do not let it dry. Take another cloth and wipe the glass door, making sure that you remove all of the WD-40. If the stains are really bad and have been there for some time, you may have to repeat this step, but you can be sure that this will work and you will have a squeaky clean door in no time at all.
Other solutions that may work, though the above one is the best, are vinegar solutions, rubbing alcohol and the detergent that you use to wash your dishes with. Dilute all of these and put the solution into a spray bottle. Spray onto your door and wipe off. These can be effective if the spots and streaks have not been on the glass for very long.