I will keep you informed suggests that it will be a continuous action. (I would not put asap in this sentance) I will inform you asap suggests you will tell them as soon as possible, so possibly after something is completed, not a continuous action. Hope this helps

To 'keep someone informed' implies a longer period. This suggests that you will tell someone every time you receive some news about a particular subject. It's similar to 'keep you up to date' or 'keep you updated', and suggests that this is a continuous and ongoing process. If you 'inform' someone, this is a single act of giving them one piece of information on one occasion. It's a more formal/businesslike word for 'tell'.