If you have ever seen the movie "The Terminator" you can understand the meaning of relentless.
"That Terminator is out there. It can't be reasoned with, it can't be bargained with...it doesn't feel pity of remorse or fear...and it absolutely will not stop. Ever. Until you are dead."
It will just keep trying until it succeeds, no matter what obstacles get in its way. No matter how much it gets damaged.
Persistent is less intense. It just means making constant effort over time. It doesn't have the same implication of sacrificing everything to accomplish a goal.