Although on the face of it, both words seem to mean resolution or endurance, I don't think they are always interchangeable. The first seems to have positive connotations and the second negative ones.

Perseverence: more dedication and diligence
Persistence: more pertinacity, tenacity

I wouldn't mind someone saying in a reference that I persevered; I might not be so happy if they said I persisted.

Peace negotiators persevere.
Hawkers, flies and pit-bulls persist.

This is only my take on this; others may have different ideas.

I’ve just had a hard think about that. It’s difficult to make out a difference, but it occurred to me that perserverance could be seen as more of a traditional virtue. What do other LEOs think?

Edith: #1 sums it up very nicely.

Just noticed my typo: of course it should be perseverance

Thanks, tomtom! On second thought I do agree with your distinction.

I can be persistant even if I am not faced with great difficulties. Perseverance suggests that I persist in the face of great difficulty.

from dictionary.com

perseverance: 1. steady persistence in a course of action, a purpose, a state, etc., especially in spite of difficulties, obstacles, or discouragement.

I won't (necessarily) say the two words are (necessarily) interchangeable in every conceivable sentence. But I don't see "persistence" as being inherently negative at all.

It's true that "persevere" seems (almost always) to be used in the pursuit of something noble or worthy--and is usually not used if the goal is bad.

But "persist"--though it can be used in negative contexts ("He persisted till he had eaten the entire cake, leaving none for anyone else")--can indeed be (and often is) used as meaning the pursuit of positive ends:

The woman persisted in painting the huge apartment.

He persisted until he graduated from medical school.

The policeman persisted until he found the arsonist.

The clergyman persisted in working with the young man, who finally saw the light and repented.

The woman persisted in yelling at the dentist, who was pulling the wrong tooth!