A few quotes to get started:
Romans 8: 24-25 (NIV 2011) "For in this hope we were saved. But hope that is seen is no hope at all. Who hopes for what they already have? But if we hope for what we do not yet have, we wait for it patiently."
Yoda (Star Wars) "Fear leads to anger. Anger leads to hate. Hate leads to suffering."
Romans 5: 3-4 (NIV 2011) "Not only so, but we also glory in our sufferings, because we know that suffering produces perseverance; perseverance, character; and character, hope.
(The Yoda quote is up there as an afterthought, to tie into the quote from Romans 5. If you connect the two, you find that fear gives us hope, but not without suffering as well. Then again, Yoda and Paul aren't quite talking about the same kind of suffering, since the former is self inflicted and the latter is given from others. I just like how the two seemingly connect to bring together two opposite ideas.)
You probably know what patience is by this point in your life. It's that thing that we all want and often lose, yet we don't want to be put into a situation in which we get more. It requires waiting without complaining for the arrival of our hopes.
Perseverance is a fairly long word, and isn't used as frequently in vocabulary as patience. Along with being harder to spell, it usually gives us more grief than patience, since perseverance comes from enduring some sort of suffering.
Imagine a runner in a race. The runner's fans need patience as they wait at the finish line because they're anticipating the arrival of the runner. The runner, however, needs perseverance to keep running despite the pain.
When you think about it, the difference between the two is subtle. Patience is waiting for a later gift, and perseverance is waiting for the removal of a current problem. I find this somewhat similar to the connection between mercy (removal of the harm you deserve) and grace (gift of good you don't deserve).