Why does a sign difference between space and time lead to time that only flows forward? Ever since special relativity we've had this equation that puts time and space on an equal footing:
$$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2.$$
But they're obviously not equivalent, because there's a sign difference between space and time.
Question: how does a relative sign difference lead to a situation where time only flows forward and never backward? We can move back and forth in space, so why does the negative sign mean we can't move back and forth in time? It sounds like something I should know, yet I don't - the only thing I can see is, $dt$ could be positive or negative (corresponding to forwards and backwards in time), but after being squared that sign difference disappears so nothing changes.
Related questions: What grounds the difference between space and time?, What is time, does it flow, and if so what defines its direction?
 However I'm phrasing this question from a relativity viewpoint, not thermodynamics.
 A: 
Why does a sign difference between space and time lead to time that only flows forward?

It doesn't. Spacetimes don't even have to be time-orientable. This is similar to the idea that a Mobious strip is not an orientable surface. So you can have a metric with a $-+++$ signature but no direction you could define for time to flow, even if you got to set up the thermodynamics however you liked.
A: The minus sign does not imply that time flows only in one direction. This is seen if we define the forward direction of time to be the direction in which entropy increases. 
You cannot derive the $2^{nd}$ law of thermodynamics based on the fact that there is a minus sign on time and not the spatial parts. Indeed, this is a matter of convention as we could have put the minus sign on the spatial parts and a + on the time component. 
However: I don't think this is the heart of your question. The minus sign is due to the fact that space and time are not separate entities but are part of one vector space called spacetime or minkowski space. The relative minus sign tells us about how they are connected. Namely, it tells us about the geometry of spacetime. 
For a physical argument to why this is the case, I refer you to this answer by @Dvij Mankad.
A: 
We can move back and forth in space, so why does the negative sign mean we can't move back and forth in time?

As illustrated in the answer by user4552 and acknowledged in other answers, that relative sign doesn't by itself determine which is future and which is past. But as the answer by Dale explains, it does mean that we can't "move back and forth in time," assuming that the spacetime is globally hyperbolic (which excludes examples like the one in user4552's answer). A spacetime is called globally hyperbolic if it has a spacelike hypersurface through which every timelike curve passes exactly once (a Cauchy surface) [1][2]. This ensures that we can choose which half of every light-cone is "future" and which is "past," in a way that is consistent and smooth throughout the spacetime.
For an explicit proof that "turning around in time" is impossible, in the special case of ordinary flat spacetime, see the appendix of this post: https://physics.stackexchange.com/a/442841.

References:
[1] Pages 39, 44, and 48 in Penrose (1972), "Techniques of Differential Topology in Relativity," Society for Industrial and Applied Mathematics, http://www.kfki.hu/~iracz/sgimp/cikkek/cenzor/Penrose_todtir.pdf
[2] Page 4 in Sanchez (2005), "Causal hierarchy of spacetimes, temporal functions and smoothness of Geroch's splitting. A revision," http://arxiv.org/abs/gr-qc/0411143v2
A: The sign that appears in the metric or line element, i.e. in
$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$
does establish a difference between space and time, but it does not, on its own, contain all of the physics related to time. For one thing, it does not determine which direction is future and which is past. That direction is established by other considerations such as entropy increase. The other main ingredient here is the claim that worldlines are timelike not spacelike. This really amounts to a statement about conservation laws. We identify a sequence of events along a certain line in spacetime as a sequence associated with one particular entity, such as a particle or a body, because there is something in common at the events: a certain amount of electric charge, for example, or energy and momentum. So we say we have a particle (or larger body) and that is its worldline. A sequence of events along a spacelike line, on the other hand, often doesn't show that kind of common property, so we don't find it helpful to suggest that the same entity was present at all the events. As you see, we are getting quite close to metaphysics here. 
A: 
how does a relative sign difference lead to a situation where time only flows forward and never backward? We can move back and forth in space, so why does the negative sign mean we can't move back and forth in time?

It is not only the sign difference, but also the fact that there is only one dimension of time while there are multiple dimensions of space. Because there is only a single dimension of time a surface of constant proper time forms a hyperboloid of two sheets. One sheet is future times and the other sheet is past times, so there is no way to smoothly transform a future time into a past time. Future and past are geometrically distinct. 
In contrast, because there are three spacelike axes a surface of constant proper distance forms a hyperboloid of one sheet. So you can smoothly transform up into down and so forth. Different spacelike directions are not geometrically distinct. 
A: The designation "space-time", suggesting that there are 3 space and one time dimensions, is somewhat misleading; we should rather speak of 4 space dimensions, which together allow time (or movement, energy) to exist! Time is the measure of movement, telling us how fast a determined movement, e.g. the rotation of the earth, is, compared to another movement, e.g. the physicist's clock. 
That our Universe is 4-dimensional is obvious from Albert Einstein's mass-energy equivalence $ E = mc^2 $ and the relativistic energy invariant $ E^2/c^2 - p⃗^2 = m_0^2 c^2 $. Measuring distance in light-seconds instead of meters, the speed-of-light c becomes =1, and we get the simple formula:
$ E^2 = m^2 = m_0^2 + p⃗^2 = m_0^2 + p_1^2 + p_2^2 + p_3^2 $ 
From this formula it is obvious that the rest mass $m_0$ is the fourth component of the momentum (or movement) vector, and that the energy or mass is the total length (absolute value) of the momentum vector.
Mass is energy, mass is movement (of something more basic, certainly)!
Now, how is movement (or time) possible after all in the Universe? Despite of some Greek philosopher's denial of the possibility of movement, Leonhard Euler, a 18th century Swiss mathematician working in St Peterburg (Russia) found a surprising identity, stating that a sum of four squares can always be written as the product of two sums of each four squares. (This holds also for sums of two squares, and for sums of eight squares, but for nothing more - A. Hurwitz, 1895)
Applying this formula to our momentum vector, we can write:
$ (m_0^2 + p_1^2 + p_2^2 + p_3^2) =  (r_0^2 + r_1^2 + r_2^2 + r_3^2)(M_0^2 + P_1^2 + P_2^2 + P_3^2) $
wherein the components are (proof by algebraic evaluation):
$ m_0 = (r_0M_0 - r_1P_1 - r_2P_2 - r_3P_3) $
$ p_1 = (r_0P_1 + r_1M_0 + r_2P_3 - r_3P_2) $
$ p_2 = (r_0P_2 - r_1P_3 + r_2M_0 + r_3P_1) $
$ p_3 = (r_0P_3 + r_1P_2 - r_2P_1 + r_3M_0) $
Now, let's suppose that the vector 
$ P⃗ = (M_0, P_1, P_2, P_3) $ is a previous physical state, and
the vector $ R⃗ = (r_0, r_1, r_2, r_3) $ represents a physical process, transforming vector $ P⃗ $ into vector $ p⃗ = (m_0, p_1, p_2, p_3) $, and let's further assume that process $ R⃗ $ does not change the total energy of the system, i.e. $ (r_0^2 + r_1^2 + r_2^2 + r_3^2 ) = 1 $ , then the equation:
$ p⃗  =  R⃗ * P⃗ $ , wherein * is the multiplication as defined above, or, given the bilinearity of Euler's 4-squares identity, also the equation:
$ p⃗ = p1⃗ + p2⃗  =  R⃗ * (P1⃗ + P2⃗) = R⃗  * P ⃗ $
describe the movement or evolution of a system $ P⃗ $ or of
a system $ (P1⃗ + P2⃗) $ under the influence of a physical process $  R⃗ $, wherein the total energy is conserved. 
The concept of time in physics is linked to such processes, implying and requireing all 4 dimensions of 4-space. Rest mass $ m_0$ itself is such a process!
