Is Time Dilation in GR Simply a Consequence of Curved Space? My understanding of time dilation in General Relativity is that it is the consequence of particles traveling at the speed of causality following longer, curved paths. Since it takes longer for two points in space to interact due to the increase of their separation distance, time slows down. Is this all that time dilation is in GR, a consequence of the curvature of space? Or is their some other reason the rate of time changes?
 A: I'll assume that you already have an understanding of the "relative velocity" time dilation as is found in special relativity.
No, gravitational time dilation isn't caused by the curvature of spacetime.  Instead, gravitational time dilation is due to using a non-inertial frame of reference.  Gravitational time dilation is more of an issue in a curved spacetime, because if the spacetime is curved, then it isn't possible to use one inertial frame of reference to cover all of spacetime, like it's possible to do in a flat spacetime.  But "gravitational" time dilation will also show up on a flat spacetime, if you choose to use a non-inertial frame of reference.
Gravitational time dilation is fundamentally the same thing as relative velocity time dilation.  Whether the time dilation between two clocks as of two given events looks like "gravitational" time dilation or "relative velocity" time dilation, or some combination of the two, depends entirely on which coordinate system(s) you choose to use.  If you use a coordinate system in which the spatial coordinates of each clock remains constant, the time dilation will look like purely "gravitational" time dilation.   But if you're careful to use only inertial frames of reference in your analysis, the time dilation will look like purely "relative velocity" time dilation.
As an example, consider two clocks, one sitting on the roof of a skyscraper of height $h$, and the other sitting on the ground next to the skyscraper.  As developed in the gravitational time dilation wikipedia article, in the (non-inertial) frame of reference used in which the two clocks remain at the same spatial coordinates, the rate of the clock on the ground is slower due to gravitational time dilation by a factor
$$T_d=1+\frac{gh}{c^2} ,$$
where $g$ is the acceleration due to gravity at the Earth's surface, which is assumed to be close enough to being constant at the altitudes involved, and it's assumed that $gh << c^2$.
However, you could also analyze the same situation using an inertial frame of reference.  Synchronize a third clock to the clock on the roof, and let it drop from the roof.  While the third clock is in free fall, a comoving frame of reference right around the clock is an inertial frame of reference.  At the moment when the third clock lands on the clock on the ground, from the perspective of the third clock's comoving inertial frame of reference, the clock on the ground is moving at a speed of roughly
$$v=\sqrt{2gh}, $$
(neglecting air resistance), so the clock on the ground is measured to be running slow due to relative velocity time dilation, by a factor of
$$\gamma=\frac{1}{\sqrt{1+\frac{v^2}{c^2}}} = \frac{1}{\sqrt{1+\frac{2gh}{c^2}}}\approx  1+\frac{gh}{c^2} .$$
That is, as measured in the third clock's inertial frame of reference, the clock on the ground as of when it collided with the third clock is running slower than the clock on the roof as of when the third clock was dropped, due purely to relative velocity time dilation.  The calculation using the non-inertial frame of reference agrees as to what the speed ratio between the two clocks is as of those two events, but the speed ratio is considered to be purely a matter of gravitational time dilation.  It's the same ratio describing the same physical events; the two different perspectives lead to only a meaningless disagreement about "what kind" of time dilation occurred.
A: They are separate phenomena. But time dilation is inherent in both GR and SR because "time is just a coordinate". In SR, time dilation happened because different frames cannot agree on what it means for clocks to be synchronized.
In GR it is actually quite similar, since GR is kind of like "SR sewn into a curved spacetime fabric". Matter and energy curves otherwise flat spacetime. And by looking at the metric of spacetime, one can deduce gravitational-based time dilation (which is different from special relativity). One typical example is the so-called gravitational time dilation: in Newtonian language, a clock at lower gravitational potential $\phi$ (i.e. stronger gravitational field) ticks slower than a clock at higher $\phi$. This happens to our GPS - GPS must account for slight difference in time flow in orbit compared to surface of the Earth on top of accounting for SR time dilation to remain accurate.
