Why does an Alcubierre drive, travelling FTL, violate causality, if the universe expanding FTL doesn't? An Alcubierre drive seems to be plausible as a means to travel faster than light, because

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*it doesn't move the object itself, but the space around it.

*it's said that matter and information can't move through space faster than light, but space itself can.

However, it turns out that an Alcubierre drive still can't achieve superluminal speed without violating causality, because:

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*the limit of speed of light doesn't apply only to matter and information, but to anything that moves from one point to another in space.

*anything that moves faster than light also makes information that comes out of it travel back in time, which violates causality.

*since the warp bubble itself moves faster than light within the rest of the universe, an Alcubierre drive will violate causality as it moves faster than light.

But if this is true, why can the universe expand faster than light, without time travel and therefore violation of causality happening? What makes it different to a warp bubble?
 A: There are two fundamental differences between these two cases.
First, an Alcubierre warp tube is supposed to be something that you make. The problem is that if you can make one of them, and the laws of physics are Poincaré (Lorentz+translation) invariant, then you can make two of them in a configuration that violates causality, as Kevin Kostlan said.
This is a nonissue for the large-scale structure of spacetime, because you can't make two universes, or even one universe. Its shape is (by assumption) out of your control, and it has, as a matter of fact, no closed timelike curves.
Second, the expansion of the universe is, in a certain sense, faster than $c\triangleq 299{,}792{,}458\text{ m/s}$, but it isn't faster than light. A galaxy distant enough to have a recession speed larger than $c$ won't outrun a receding beam of light at the same distance, because the light's recession speed is even larger. $c$ is only the speed of light in inertial coordinates, and the standard cosmological coordinates aren't inertial.
If you go through an Alcubierre tube – even a unique one made by God so there are no causality issues – then you do outpace light that travels between the endpoints outside the tube. Nothing analogous to that ever happens in cosmology.
A: In any curved spacetime we can still talk about local reference frames that are small enough scale we can ignore the curvature. We also can ask if there are closed timelike curves (CTC) which basically is asking whether we can time-travel to our past selves. CTCs are strongly thought to be impossible in reality.
The universe is thought to be spatially flat, but the spacetime as a whole is curved. CTC's are impossible: at each point in spacetime you have an "age of the universe". To be precise, this is maximum path-length (proper time) a geodesic could have between the big-bang singularity and said point. Any time-like or light-like curve is moving in the direction of increasing age of the universe; this is just as strong a concept of "future" and "past" as in flat spacetime.
With a single warp-drive you don't have CTC's. But you can still get CTC's with multiple warp-drives. Suppose you build a warp-drive on Earth and send it out into space. You start with an (almost) flat initial-condition and then generate a strongly curved spacetime (your warp bubble). Starting from a flat spacetime (or for very large scales from the spacetime of the universe), is much more physically realistic than starting from any other spacetime. You have to make your weird and wonderful curvature from an "empty canvas" !
With a warp-bubble, the highly curved spacetime is on a small scale. This allows us to glue two bubble spacetimes together so long as the ships don't get very close to each-other. If we consider two Earths, moving relative to each-other, that each make a warp-drive, we can set up the system to generate CTCs. This is one reason we suspect this to be impossible.
There is another reason to suspect making warp drives is impossible: Geodesics would have to diverge in some region, which is an anti-gravity effect. Neither matter nor light can make anti-gravity (antimatter has positive mass just like matter). The "attractive gravity only" rule is more precisely defined as an energy condition and at least one of these is violated by warp drives. Violating certain energy conditions would make the speed of sound faster than light which also allows for time-travel paradoxes.
In general, no known solution with CTC's is physically realistic. They either involve infinitely large systems that cannot be setup from an "empty canvas" or violations of energy conditions. For example, the Kerr metric concentrates it's energy condition violation in it's singularity. Real black holes are thought to lack this feature and be much deadlier instead.
