Animals with larger brains are more aware of time passing. It is probably this awareness that causes stress. You are likely to relate your own pressure to time. In fact, a high proportion of pressure arises from the deadlines we are set or those we set ourselves. Only, I think, one of perception.
Pressure suggests power, strength, aggression, dynamism even.
Stress implies weakness, defensiveness, failure. The two words are different sides of the same coin. Which word applies depends on how you deal with the pressure or stress. I am a perfectionist. That means that I set standards which are high and put pressure on myself to achieve those standards.