In the illustration below, the software that rendered it, CorelXara 2, has introduced an anomaly in the horizontal and vertical parts of the circle that conveniently illustrates this perception of varying thickness. The horizontal parts of the circle, which has a 1-pixel outline, appear to be too thin because the outline is represented at this point by just single pixels. However, the angled portions of the circle are represented by two pixels. And the vertical section of the circle here actually looks to be of the correct thickness because more intermediate pixels have been inserted along almost the whole length of that section, so it appears to be two pixels thick. For thicker lines, this varying-thickness effect is less pronounced, as the extra pixel required is a smaller percentage of the overall thickness of the line.
Most software also handles text objects and graphical objects using two different algorithms.
This is because text has its own inherent problems due to the complex shapes in the glyphs, plus the sheer number of character objects present. Thus, algorithms optimised for handling text and fonts antialias those objects, while more general algorithms are applied to other objects. In principle there isn't a problem with this, however the results from a piece of text should be identical whether that text is still in text form, or whether it has been converted to curves. Also, a font technology known as 'hinting' can affect these results. Hinting is extra information embedded in a font that describes to the font-rendering engine how to display that font at very small screen sizes using just black pixels, i.e. no antialiasing.