"Persistence" is used in the sense of "stubborn" (obstinate). There could be a slight negative connotation. For example:
The telemarketers persistently called us.
We don't say they "perseverantly" called us. "Persistent" is used when there continuing action, and the purpose is either bad, or neutral, and sometimes(rarely) good.
"Perseverance", is used when the purpose is good. That is, you're doing something to help someone, or to achieve a desirable goal, such as climbing Mt. Everest.
Depending on how you feel on something, you could use "perseverance"/"Persistence". For example, the telemarketer calling you, could annoy you, and you could call him "obstinately persistent". But to the marketer, you might be to him, a tough customer, and he might seem to himself "bravely perseverant in trying to persuade this tough customer."
As you can see, it gives a different "feel" about the thing described.