The issue here has to do with how the browsers fit the mathematics to the pixels of the screen. For characters in a font, most browsers use antialiasing to help perform "sub-pixel" positioning, allowing characters to appear to be positioned anywhere, without having to line up with the pixels exactly. The horizontal and vertical lines, however, are not characters from a font (they are actually implemented as borders of elements with given dimensions), and so they are not anti-aliased. If a horizontal line is asked to be placed so that its has to be divided between two pixels, the browser has to decide which pixels to use for it. If it is mostly in the upper pixel, that pixel is used; if mostly in the lower one, that pixel is used; but if it is nearly half in each, what to do then? Some browsers show both.
I believe that is what you are seeing here. If you inspect the DOM elements using Firebug or some other similar tool, you will see that the heights of all the horizontal lines are exactly the same in the CSS that controls them. The different sizes are purely the result of the browser attempting to render those sizes in the pixel grid that it has available. The result is very much dependent on the position of the lines compared to the pixel grid of the screen (which is why only some of the lines are thicker). If you zoom the page in or out, you will get different ones that are too thick. I've even seen them change as the page is scrolled in some browsers.
I have looked into this issue before, and there seems to be no easy answer for it. The situation is particularly delicate for sizes that are near one pixel, as getting an extra pixel in that case is very apparent. On the other hand, if the size of the line is made less than a pixel in order to reduce the chances of the extra pixel, some browsers will drop the line entirely (thinking it is less than a pixel so shouldn't be shown). In general, MathJax makes the lines slightly larger to avoid the disappearing lines, at the expense of some wider lines like the ones you are seeing. Firefox is particularly prone to this excess size, and IE to both the dropping and enlarging of the lines.