The present simple, "I look forward to..." is quite formal and 'detached'. We use it as a polite closing formula in business and legal letters. It's basically a way of taking the correspondence forwards and saying what you expect from the other person. For example "I look forward to your prompt reply" is a businesslike way of saying you want the receiver to answer soon, and "I look forward to a timely resolution of this matter" means that you expect a problem to be solved. The present continuous, "I'm looking forward to ..", is more informal and personal. If you say, for example, "I'm looking forward to meeting you next week" that's a genuine statement indicating a positive feeling. You can use this in any context, both in conversation and in writing.

The difference is in the amount of emotion and familiarity they imply. We use 'I look forward to XXX' for work and business because it means 'I will be pleased when XXX happens in the future'. e.g. I look forward to meeting you next week. We use 'I'm looking forward to' usually for friends/family/social occasions because it means 'I will feel really excited/happy when this happens'. e.g. I'm looking forward to going to the cinema on Friday night. It would usually seem a bit strange to say to a client 'I am really excited about meeting you next week' - too emotional. However, say you had just won a business project and you wanted to let the client know how happy about it you were, you might say 'We are really looking forward to working with you on this project'.