It's important to realize that the Apollo fire occurred in a pure oxygen atmosphere at high pressure. As I recall, they were testing at more than 1 atmospheres (over 14.7 psi). For pure oxygen, a very dangerous mix.

While in space, the atmosphere in Apollo was pure oxygen, but at a much lower pressure (about 5 psi). At these lower pressures, pure oxygen isn't nearly the combustion hazard it is at higher pressures.

Running at these low pressures allowed the astronauts to get in and out of spacesuits quickly, without problems, since the suits operated with oxygen at the same low pressure.

At present, the Space Shuttle and Space Station operate using a normal Earth atmosphere, a mix of oxygen and nitrogen at 14 psi. Unfortunately, since current space suits still operate on pure oxygen at low pressure (5 psi), this makes EVA very difficult. Astronauts must go through a prebreathing sequence to flush nitrogen out of their systems. Going directly into the low-pressure suits could actually result in decompression sickness, or the "bends."

Some efforts have been made to design suits that can operate at higher pressures (8 psi or more) to remove this problem. There have been several successful ideas in this field, including the Ames "hard suit," but NASA never took the ideas into production.