Does velocity or acceleration cause time dilation? What causes time dilation?  Acceleration or velocity?
I've seen multiple comments on this forum that assert velocity is the cause, but that doesn't seem right to me.  You can't have velocity without acceleration.  It's the inertial force with acceleration that breaks the symmetry.  My understanding is that the asymmetry is where the inertial frame changes.  Measuring time between two objects with different inertial frames is where you have time dilation.  When the acceleration ends, the object is effectively at rest in a new inertial frame and has velocity relative to another object in the original inertial frame.
In other words, acceleration (changing reference frames) is the cause...velocity and time dilation is the effect.
Am I right about this?  If there are flaws in my logic I'd like to find and correct them.
 A: Let me present a slightly different perspective to Alfred's answer, although I'm basically saying the same thing.
I suspect you've got hung up on the idea that velocity causes the relativistic effects like time dilation, but the underlying cause is something different. All the weird effects in SR are caused by a fundamental symmetry of spacetime, which is that the proper time, $\tau$, is an invarient i.e. it is the same for all users.
Suppose we take any two spacetime points $(t_1, x_1, y_1, z_1)$ and $(t_2, x_2, y_2, z_2)$ then the 4-vector joining them is $(\Delta t, \Delta x, \Delta y, \Delta z)$, where $\Delta t = t_2 - t_1$ and so on. The proper time is defined as:
$$ c^2\Delta\tau^2 = c^2\Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2 $$
Or more concisely:
$$ \Delta\tau^2 = \eta_{ab} \Delta x^a \Delta x^b $$
where $\eta$ is the Minkowski metric and we adopt the usual convention of setting $c = 1$.
The quantity $\Delta\tau$ is an invarient and all observers looking at the two spacetime points will measure the same value for $\Delta\tau$ regardless of where they are or how fast they are moving or accelerating.
To see why velocity has an effect on time and space consider this:
Start in your rest frame and measure some infinitesimal time interval $dt$ with your stopwatch. In your frame the interval between starting and stopping the stopwatch is just $(dt, 0, 0, 0)$ and therefore the proper time $d\tau$ is just equal to your stopwatch time $dt$. (I've sneakily switched from $\Delta$ to $d$ because if you're considering accelerated frames you need toi integrate $d\tau$ to get the $\Delta\tau$)
Now consider some frame that moves between you starting and stopping the stopwatch. It doesn't matter whether the frame moves at constant velocity or whether it accelerates in some manner. Because in this frame the stopwatch has moved while it was timing the interval will be of the form $(dt', dx', dy', dz')$ i.e. in this frame the spatial parts of the interval won't be zero. But we require that $d\tau' = d\tau$ because the proper time is an invarient. Equating the two proper times gives us:
$$ dt^2 = dt'^2 - dx'^2 - dy'^2 -dz'^2 $$
And because the spatial terms are nonzero this means $dt^2 < dt'^2$ i.e. the times in the two frames are different and we have time dilation.
Note that I haven't restricted how the two frames have moved relative to each other, only that they have moved. So you can't say the time dilation is due to velocity or due to acceleration, just that it's due to relative motion.
A: We need to untangle this a bit but first:  the cause of time dilation is the geometry of spacetime which is such that there is an invariant speed c.
Now, remember that velocity or speed is not a property of an object; there is no absolute rest.
Further, consider the case of three objects in uniform relative motion with respect to each other.
If I choose one of those objects and then ask you "what is the relative velocity of this object?", the only proper response you could give is "velocity relative to which of the other objects?"
So, we can't speak of the relative motion of an object but rather the relative motion of a pair of objects.
What we can say is that, for two objects in relative uniform motion with respect to each other, the other object's clock runs slow according to each object's own clock.  This is called relative velocity time dilation.
It is important to realize that in the case of relative time dilation, the two relatively and uniformly moving clocks are spatially separated except at one event.  Comparing the readings of the two clocks when spatially separated requires additional spatially separated clocks synchronized and stationary in their respective object's frame of reference
But, we find that clocks synchronized in one object's frame are not synchronized in the other relatively moving object's frame.  Thus, the relative velocity time dilation is symmetric without contradiction.  We can't say that one or the other clock is absolutely running slower.
Now, within the context of Special Relativity, acceleration is absolute, i.e., an object's accelerometer either reads 0 or it doesn't.
And, a fundamental result in SR is that a clock along an accelerated worldline through two events in spacetime records less elapsed time between those events than a clock along an unaccelerated world line through the same two events.
Since, in this case, an accelerated clock and an unaccelerated clock are co-located at two different events, the two clocks can be directly compared and, in this case, the time dilation is absolute - the accelerated clock absolutely shows less elapsed time than the unaccelerated clock.  
A: Consider this. 
The twin paradox, but with a twist.
The twin that accelerates is the one that is younger on return. Statement one
It doesnt matter which direction the accelerating twin goes. Ie if she leaves from the equator and goes due north from the earth and returns, it doesnt make any difference if she had gone due south and returned.  Statement two
But what if there are triplets? one (a) goes due north, one (b)  due south  and one  (c) stays on the earth. ( a and b's  acceleration and velocity are identical except the one is opposite the other in sign (+/-) and we can by convention decide to make the + acceleration away from the earth in a due north direction)
so when  a and b are back they are exactly the same age, but younger than c.  If statement one and two are correct, this must also be correct - Statement 3
Even though  the velocity difference between a and b was  greater at all times than the velocty difference between a and c was or between  b and c was.
Therefore it is not the velocity or even relative velocity per se, but the acceleration which causes the clocks to slow down such that when a certain RELATIVE velocity is reached, the clock travelling at the faster velocity compared to the ORIGINAL velocity is going slower than the ORIGINAL clock which has not accelerated. statement 4
Note. By different relative velocities, the two clocks as observed by different moving observes will not tick at the same rate. 
Now the question is, between a and b , whose clock is going faster or slower?  the answer is that if a and b leave from the same point at the same time, travel exactly the same distance and get back to the same orignial point at the same time and are the same age when they get back, both their clocks must have ticked at the same OVERALL AVERAGED dilated rate compared to the original clock, but not necessarily always syncronously. If A would observe Bs clock during the travelling and B would observe A's clock, then there are two components to the time dilation either observe. One is the time dilation due to  relative velocity, and the other  is the time dilation (or contraction) due to the relative acceleration ( deceleration) (which is not the same as the absolute acceleration) These two different dilations (and contraction) effects exactly cancel out, so that A and B arrive back the same age. statement 5
Now as c stayed on earth, and was subject the whole time to an acceleration too ( ie gravity) her clock is slightly dilated too.....( as there is no difference to clocks as to what causes the acceleration  ie either gravity or rocket engine) 
This then brings the following conclusion.....
A clock that has accelerated ticks more slowly than a clock which has not accelerated so there is time dilation at a higher velocity.
So during the acceleration, the clock will start getting slower and slower.
Now as a gravitational acceleration has the same effect on clocks as a rocket powered acceleration, and as a clock in a rocket powered acceleration will get slower and slower the longer the acceleration occurs,  then a clock that exists say for 100  million years on the earth and is subject to gravity for the entire period and therefore an acceleration for 100 million years will tick more slowly now in 2015 than it did 100 million years ago.
A: In Special Relativity the time dilation is just a matter of convention in the time measurement between moving frames.
In General Relativity the time dilation is a physical phenomenon which involves a force field (either gravity or acceleration or wathever) that actually slows down the particles of a system. When particles interactions take place with lower frequency, Time (lifetime, aging etc.) is effectively slowed down.
