There is a value difference between competent and proficient: While both refer to knowing a skill, competency can refer to the bare minimum required for acceptability. Proficiency carries with it a level of mastery that is above the minimum. "Competent" lives in an either/or framework . One is either competent or incompetent. Proficiency continues: "I am gaining proficiency in reading". One can be competent without being spectacular, but continues to grow in his or her proficiency. Accomplished refers to having a successful track record of achievements.

Yes I think they are interchangeable. I can't think of an example where one is ok and the other isn't. There may be some very subtle implications in some cases that competent means ability levels from foundational, or minimal passing, to expert, whereas proficient means only fairly skilled to expert, but overall they overlap totally I would say.