The two kinds of paradox are extremely closely related, so many times a movie which seems to deal with one actually deals with both. In fact, I've found quite a few places that consider them to be different names for the same thing.
In a predestination paradox, an event in the past creates a chain of events that leads to some second event in the future, but that second event is ultimately the cause of the first event. This is equivalent to what theoretical physicists would call a "closed time-like loop" (something that is "allowed", though extremely unlikely to be real, according to relativity.) The key here is that the action in the future only happened because the action in the past happened, but the action in the past was explicitly caused by the action in the future.
In a bootstrap paradox, an object (which could be a physical object, or a piece of information) is sent to the past, and the receipt of that object triggers a series of cause/effect, which ultimately leads to a future event whereby the same object is sent back in time. The key here is that the object/information did not come from anywhere. It exists because it always existed. (This kind of paradox is not, as far as I understand, even theoretically possible under relativity, it violates several conservation laws.)
I know that you asked for movies, but really, TV shows are a much bigger pool of information on this stuff, if for no other reason than simple quantity. A good primer on both of these kinds of paradox is Doctor Who, since it involves quite a lot of time travel. An example of each (there are tons more):
The highly popular, acclaimed episode Blink contains a bootstrap paradox: The Doctor (stuck in the past) reads a transcript of a conversation into a camera, which he arranges to be embedded on a DVD. A girl in the future watches the DVD and transcribes the conversation, then later gives the transcript to The Doctor, who takes it into the past with him to read.
The mini-episodes Time/Space and Time Crash both feature The Doctor giving information to a future version of himself; it's much more explicit in Space.
The episode Fires of Pompeii contains a (sort-of) predestination paradox. The Doctor finds himself in Pompeii just before Vesuvius erupts, and spends much of the episode trying to escape. It is later learned that he caused the eruption, in order to prevent a much worse disaster that would have resulted in the destruction of the entire Earth. Had he not done so, at the very least his companion from present-day Earth would never have existed, so he would never had taken her back to Pompeii, etc.
Many other TV Shows that deal with time travel mix the two up; the entire third season of Fringe is a giant bootstrap/predestination paradox:
Peter sent the blueprints for The Machine back into the past, along with some of it's parts, by using the machine that was built from the blueprints he sent into the past.
As far as movies, it seems to be much harder to find recent examples of this; many time-travel movies these days explicitly avoid these two kinds of paradox by changing the future (Looper, Butterfly Effect, Days of Future Past, etc.) Both of these paradoxes involve a stable timeline -- nothing changes as a result of the time traveler. Most of the examples I found are from previous decades (or based on older books).
For the predestination paradox, other than Predestination, a good example would be:
- The Time Traveler's Wife: the protagonist travels through time meeting his wife at various stages: She only falls in love with him because he time traveled to meet her as a child, but he only visited her because they were in love in the future, etc.
For the bootstrap paradox, a few that I came up with:
The Terminator series (specifically, 1 & 2): The original Terminator was sent back in time by Skynet to kill Sarah Conner; that Terminator is later found by Cyberdine, reverse-engineered, and used as the basis for Skynet. Thus, the technological information needed to build Skynet came from Skynet.
Bill and Ted's Excellent Adventure: a very small one, but they only know to call Rufus "Rufus" because they hear themselves call him that; he never actually gives his own name.
There's also a lot of implied bootstrap paradoxes in movies, but they require a bit of assumption. For example, in Star Trek IV, Scottie gives an engineer the formula for transparent aluminum. If we assume that engineer patented that formula, and the rest of the world learned it from him, that would count. However, that requires us to reject the possibility that someone else independently discovered that knowledge on their own, which is something that happens all the time in science and math.