“I’m running late” is what you would say to express that you’re going to be late to something. I have an appointment at 10, but I’m running late. I won’t get there until 10:15. “I’m late” is what you would say after you arrive. “I know I’m late, sorry, but I’m ready to get started. Shall we?”

I go to the doctor's office. My appointment is at 2 p.m. The doctor sees me at 2:30 p.m. She says "I'm sorry I'm late." I go to the doctor's office. The doctor has a busy schedule. Everything is taking longer than usual. The doctor was late for her previous appointment. The doctor is going to be late for my appointment. The doctor is going to be late for every appointment this afternoon. The receptionist says "I'm afraid the doctor is running late today." "I am late" refers to a single event. "I am running late" refers to a whole schedule of events, all of which are delayed.