Distance between two points in UTM coordinates. This question is partially about geometry of the sphere and partially about Universal Transverse Mercator coordinates. I realize the latter is not completely on-topic here, but I hope the question is still close enough to be deemed relevant.
UTM coordinates are given as a zone number together with an easting-northing pair. Given two points $(E_1,N_1)$, $(E_2,N_2)$ in the same zone is it the true that the distance "over the Earth" between the two points is well approximated by the euclidean distance between the easting-northing coordinates? I.e.:
$$\sqrt{(E_1 - E_2)^2 + (N_1 - N_2)^2}$$
 A: It depends on your margin of error, but in general I'd say yes, the approximation is quite close.
UTM projects onto a cylinder, and a cylinder is essentially flat (zero Gaussian curvature) so the formula would be accurate for points on the cylinder. Now the zones of UTM are sufficiently thin that the distance between actual surface of the earth and that imagined cylinder is not too large. Therefore the error in distance is not too large either. Obviously this depends on your idea of “not too large”: for estimating the range of an airplane or some such, it should be quite sufficient, but I wouldn't suggest navigating the plane through fog using that approximation.
If you look closer at the Wikipedia article you referenced, you will find the following statements:

By using narrow zones of $6°$ of longitude (up to $668$ km) in width, and reducing the scale factor along the central meridian to $0.9996$ (a reduction of $1:2500$), the amount of distortion is held below $1$ part in $1{,}000$ inside each zone. Distortion of scale increases to $1.0010$ at the zone boundaries along the equator.

So that gives you a quantitative idea of how well your approximation works: at worst it will fall short by $0.4‰$ or exceed the actual value by $1‰$.
A: There is another aspect which is not touched on by the previous answer, that is the altitude or elevation. 
The Euclidian UTM approximation to distance across Earth you give is actually an approximation to the distance across the surface of the geoid at that location. That needs to be scaled by 
$$
\frac{(h+R_0)}{R_0}
$$
where h is the height above the geoid (~sea level), and R0 is the radius of the Earth or ~6371 km. In most locations that correction is smaller than the simple Euclidian approximation error but could be important in some locations such as a 5000 m high plateau in the Andes where it is ~0.08%.
A: One additional aspect not addressed above is the nature of the terrain between the two points. UTM is a $2$ dimensional Cartesian coordinate system. Points on (above or under) the Earth's surface exist in a $3$-dimensional environment with hills, valleys, etc. To illustrate the simplest case, point $B$ is at a different elevation ($e$) than point $A$ with a constant slope. After calculating the planar distance ($d$) between the two points (as described above), one must then adjust for the slope such that the actual distance ($D$) is defined by the following formula:
$$D = \sqrt{( .9996d)^2 + (e_B - e_A)^2} \cdot \left(\frac{\frac{e_B+e_A}2 + R_0}{R_0}\right)$$
Where 
$d$ is the planar distance between points A and B assuming both are at sea level;
$e_B$ is the elevation of point $B$;
$e_A$ is the elevation of point $A$; and
$R_0$ is the radius of the Earth.
