Is the Sobel filter a high pass filter, and if not, what is the difference between them?
They are both highpass type filters, but used with very different intentions. 

One should immediately observe the fundamental difference that the output of unsharp masking filter is an enhanced image to be viewed by humans, whereas the output of the Sobel (edge detector) filter is not an image to be viewed by humans, but rather a description of the edges to be further analysed by a machine vision system.

Digital imitation of an unsharp masking effect does not necessarily explicitly use a highpass filter in its implementation, yet effectively the operation that it performs is similar to the operation that a highpass filter performs, therefore it's equally valid to call itself as a highpass filter too.

More specifically, unsharp masking operates by subtracting a lowpass filtered version of the image from itself. Stated in impulse responses;

$g[n_1,n_2]$ being the output image and $f[n_1,n_2]$ being the input:

$$ g[n_1,n_2] = a \cdot f(n_1,n_2) - b \cdot f_L(n_1,n_2) $$

where $ a &gt; b $ are two constants and $f_L[n_1,n_2]$ is the low pass filterd image,

$$f_L(n_1, n_2) = f(n_1,n_2) \star h_{LPF}[n_1,n_2] $$

where $h_{LPF}[n_1,n_2]$ is a lowpass filter and $\star$ denotes convolution.

Reordering the equation and noting that $$f[n_1,n_2] = f_L[n_1,n_2] + f_H[n_1,n_2]$$ we can see that 

$$g[n_1,n_2] = (a-b) f_L[n_1,n_2] + a f_H[n_1,n_2] $$

or more specifically

$$g[n_1,n_2] = f[n_1,n_2] \star ( \delta[n_1,n_2] - a h_L[n_1,n_2])$$
$$g[n_1,n_2] = f[n_1,n_2] \star h_{eH}[n_1,n_2] $$

The last variable $h_{eH}[n_1,n_2]$ is the effective highpass filter that the unsharp mask algorithm is performing. So as you can see, it effectively includes a highass filter.

Unsharp masking is a picture enhancement technique borrowed from analog film processing era, which is used to improve the apparent sharpness of an image to be viewed. Since it increases the sharpness, it's effectively a high pass filter, but not very strong. It enhances edges, by making them stronger, too, and this is how it achieves improved, percepted, sharpness effect.

Sobel filter is actually an edge detection method and as such it's a nonlinear operator. However, its internal operation involves (basically) two directional impulse responses along horizontal and vertical directions that apporixmate the gradient of the image by using two LTI high pass filters. As, such it's using high pass filters internally, but its output image is obtained by further nonlinear processing (to keep edge pixels and throw out the rest) whose frequency response character does not exist, conventionally.

Sobel filter is, effectively, a very strong highpass filter used for machine vision purposes; specifically for edge detection. Since it's a strong highpass filter, it will eliminate all the low frequency content and its output will be black background and white lines denoting the edges, in addition to noisy dots and contours. It's never used to improve the sharpness of an image.

Sobel detector is fundamentally a derivate filter; note that derivative filters are always of highpass character, whereas integration filters are of lowpass character. They (the Sobel filters) compute the approximate derivative (or Gradient in 2D application) of a function at around the pixel points. This derivative is an indicator of a possible edge.