Here is what Free Online Dictionary says about the word "to practice":
1. To do or perform habitually or customarily; make a habit of: practices courtesy in social situations.
2. To do or perform (something) repeatedly in order to acquire or polish a skill: practice a dance step.
3. To give lessons or repeated instructions to; drill: practiced the students in handwriting.
4. To work at, especially as a profession: practice law.
5. To carry out in action; observe: practices a religion piously.
6. Obsolete To plot (something evil).
1. To do or perform something habitually or repeatedly.
2. To do something repeatedly in order to acquire or polish a skill.
3. To work at a profession.
4. Archaic To intrigue or plot.
And here is the definition of the word "to rehearse":
1.a. To practice (a part in a play, for example) in preparation for a public performance.
b. To direct in rehearsal: rehearsed the orchestra.
2. To perfect or cause to perfect (an action) by repetition. See Synonyms at practice.
3.a. To retell or recite.
b. To list or enumerate: rehearsed her complaints in a letter. See Synonyms at describe.
To practice something, such as a speech, before presenting it publicly.
I think the point No. 2 is the key for the term "to practice": "To do or perform (something) repeatedly in order to acquire or polish a skill". In the word "to rehearse", No. 2 is also very similar in meaning: "To perfect or cause to perfect (an action) by repetition".
I guess in some cases these words can be used interchangibly. But to me, the word "to rehearse" has more association with No 1. a. "To practice (a part in a play, for example) in preparation for a public performance and b. To direct in rehearsal".
The biggest difference in these words for the purpose of organ playing is that when we practice, we polish something, perform something repeatedly in order to perfect it. When we rehearse, we are preparing for public performance.
What do you think? Share your thoughts in comments.