I think in BrE these two are almost interchangeable. A third choice would be "as soon as I can".
To be really pedantic, "fast" may indicate the speed I will travel to the apartment. "Soon" could indicate that I will set off immediately, without delay, even though the travel may be slow. "Quickly" may be closer in meaning to "soon".
But I still think they are all used to mean almost exactly the same things.
Student or Learner