Does gravity CAUSE the bending of spacetime, or IS gravity the bending of spacetime? In reading these discussions I often see these two different definitions assumed.  Yet they are very different.  Which is correct:  Does gravity CAUSE the bending of spacetime, or IS gravity the bending of spacetime?  Or do we not know?  Or is it just semantics?
Would, in the absence of spacetime, my apple still fall to the earth?
 A: I think the correct answer should be that what we call gravity is a fictional force which we experience due to living in an accelerated reference frame (as opposed to an inertial one). Unlike other forces, the force of gravity disappears by a coordinate change. If a person is in a falling elevator, they experience free fall, i.e. they feel like they are floating, and they would conclude there is no force of gravity acting on them. However we at the surface of the Earth would say that clearly the force of gravity is causing the elevator to plunge ever faster towards the ground.
Of course the solution to this odd state of affairs is that gravity is not a force at all. We live in a four dimensional universe with a pseudo-Riemannian geometry in which freely falling objects move along geodesics, or lines of extremal space-time distance. Because the geometry can be intrinsically curved (like the surface of a sphere), those geodesics are not what we think of as straight lines. The person inside the elevator moves along a geodesic, while we on the surface of the Earth are accelerated and do not move along a geodesic. The space-time paths (or worldines) of the elevator and the ground underneath it are not straight lines, and so they intersect at some point. That intersection is the point in space-time at which the elevator hits the ground.
One way to think of this is to consider two ants walking along lines of longitude on a globe. Lines of longitude are great circles, and are geodesics of the sphere. The two ants start at the equator on different lines of longitude both heading due north at the same speed. Their paths are initially parallel to each other, but as they move along the curved surface the distance between them shrinks until they eventually collide at the North Pole. It appears as though there is a force which is pulling them together, but in fact the force is fictitious, the reason they got closer is because on the sphere the geodesics converge and cross each other, unlike in flat space where the geodesics are straight lines which never cross. If the globe is very large, the ants will never know that they are moving on a curved surface, and so would conclude that there must be some force which attracts them. This is the fundamental picture for how "gravity" works from the perspective of General Relativity.
Now to your question, the difference is subtle. While what we refer to as "gravity" is subject to semantics, there is something more profound going on. General Relativity is usually referred to as a "theory of gravity", in which case we can think of the answer as the latter: by definition, gravity is the bending of space-time. On the other hand if we think of gravity as a force, the apparent force of gravity is essentially caused by the fact that space-time is curved. But we can essentially take this logic in circles if we think too much about it, it all depends on what we define "gravity" to be.
But deeper than this is the question of what causes gravity? In classical mechanics we are told that  gravity is caused by mass, in the sense that massive bodies have a gravitational field which causes them to attract. But we know that's not the right picture. So to generalize your question, is spacetime curvature caused by mass? In some sense yes, in some sense no. Einstein's equation reads
$$G_{\mu\nu} = \kappa T_{\mu\nu}$$
where $\kappa$ is a constant, the tensor $G_{\mu\nu}$ is a function of the metric, which encodes the curvature of spacetime, and $T_{\mu\nu}$ is the stress-energy tensor which encodes the matter/energy content of the universe.
Because the theory of General Relativity is fundamentally four dimensional, and there is no preferred direction to call "time", we must essentially solve Einstein's equation "all at once". Clearly the matter content of the universe will determine the curvature of the universe, while the curvature of the universe will tell the matter how to move. So you have a sort of chicken and egg problem: matter tells space how to bend and space tells matter how to move.
There is a Hamiltonian (i.e initial value) formalism for GR which works for globally hyperbolic spacetimes (that is, it is not valid for all possible spacetimes). It is called the ADM formalism (named after Arnowitt, Deser, and Misner). It does allow one to set up initial conditions for a spacetime (initial curvature and matter/energy state) and compute the evolution of that spacetime and its matter content over "time" in a way that is generally covariant (does not violate relativity of observers).  But this still does not separate the inherent link between space-time curvature and matter/energy content.
As an interesting related question, one could ask whether a massive particle moving through space can interact with itself gravitationally? That is, the mass of the particle distorts space-time and therefore alters its trajectory. There is a similar question at the end of Jackson's "Classical Electrodynamics" regarding accelerating charged particles interacting with their own radiation. I believe his conclusion is that such processes are not really considered because they would create such small corrections. In the context of GR, I would guess such questions fall in the realm of Quantum Gravity.
As to your last question, perhaps you meant "in the absence of space-time curvature". In which case the answer is no, the apple would not fall, all objects would move in straight space-time paths which never intersect and so would always remain at the same distance from each other.
A: In addition to P. G. A.'s answer:

Would, in the absence of spacetime, my apple still fall to the earth?

In the absence of spacetime there would be no you, no apple, no fall, no to and no earth. Spacetime is the basic framework in which the universe plays. It is the most basic known fabric of the universe.
A: Once again I feel like all answers are missing a crucial contextual consideration: Gravity as a force is an aspect of a model which allows us to predict events/observations. The curved spacetime of general relativity is another aspect (consequence) of a model which allows us to predict events/observations. Some of the answers write that gravity does not exist, but in the same manner spacetime does not exist. Both are valuable aspects of different models (and thus both do exist as part of those models). The Newtonian model is generally the most useful model we have found so far, but it's inaccurate for certain cases. The "Einsteinian model" is a lot more accurate, but due to its complexity a lot less useful. 
Physics can never explain 'why' something happens, it can "only" (very useful, but still "only") create a model which allows us to predict future events. So asking whether gravity causes the bending of spacetime or whether gravity is the bending of spacetime is completely a question of semantics. The highest voted answer by Kai tries to discuss "what causes gravity?" which is a fundamentally meaningless question in physics. The question only gains meaning within a model, at which point the question isn't anymore "what causes gravity", but "how does gravity relate to other aspects of our model?"... which has essentially nothing to do with the real world question of what causes gravity.
So to answer your final question

Would, in the absence of spacetime, my apple still fall to the earth?

Spacetime is no more no less than formulas in a model. They aren't reality, they just model (a simplified) reality. This might sound like meaningless semantics, but what I am trying to communicate is that the question itself is meaningless. General relativity does not work without the spacetime aspect (as it's a natural consequence of it), so in that case the apple would not even exist. Newtonian physics does not have a spacetime (as a single concept), so without it the apple still falls to the ground. And in reality - as far as we know so far (!) - the apple will always fall to the ground regardless of whatever model we humans come up with.
A: 
Does gravity CAUSE the bending of spacetime?

What causes the bending of spacetime is the presence of a massive object.

IS gravity the bending of spacetime?

Yes. 

Would, in the absence of spacetime, my apple still fall to the earth?

I do not understand exactly what you mean. However, what I can tell you is that, since the spacetime is curved by the presence of the Earth, the apple will follow the geodesics and therefore it will fall to the Earth.
A: Gravity doesn't exist. Space time exists and it does not pull.
What you think of as gravity is in fact one of many correction factors that are needed to describe space time to a 3 dimensional Cartesian monkey brain.
Take an analogous example which is slightly simpler.
Remember the militant high school physics teacher who berates anyone who dares utter the words centrifugal force?
The centrifugal force is a correction factor applied when moving from an inertial frame to a rotating frame of reference.
Jimmy appears to be flung to the outside of the car, when it turns. But to the outside observer Jimmy is going straight, whilst the car is turning.
Similarly a planet when viewed locally to the spacetime is going in a straight line when orbiting the sun. For our poor monkey brains we can't imagine a straight closed loop going around the sun. So we ignore the curvature of the spacetime, and imagine a "force" and call it gravity.
A: I comment the last question:
Would, in the absence of spacetime, my apple still fall to the earth? 
The Einstein's field equations relate the geometry of spacetime to the distribution of mass and energy:
$R_{\mu\nu} -\frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu}$
where:
$c = G = 1$ natural units
$R_{\mu\nu}$ Ricci tensor
$R$ Ricci (curvature) scalar
$g_{\mu\nu}$ metric tensor
$T_{\mu\nu}$ energy-momentum tensor  
If the R.H.S. is zero everywhere, i.e. no mass or energy, the L.H.S. simply describes a flat spacetime, that is the Minkowski spacetime. The supposition In the absence of spacetime ... is not envisageable, as what shapes spacetime is the mass and energy.  
No mass and energy ---> Minkowski spacetime  
To answer the question:
1.
If no earth: the apple would not fall (no mass to curve spacetime), but would persist in its inertial state of motion.
or
2.
If earth exists: the apple would fall, as the mass of the earth would bend spacetime.
