Hi jmac,
Regarding antialiasing, vector objects are defined mathematically but they still have to be rendered on a pixel screen. While vertical and horizontal edges can be aligned with the pixel grid to minimise the effect, diagonal or curved shapes pose a problem because you will get a stepped effect when you try to represent them using pixels (see screenshot - i've zoomed in to make the pixel grid easily visible). Antialiasing helps to smooth those steps adding semi-transparent pixels to help objects blend, however this also lets the background colours from other objects bleed through their edges (see attached screenshot - bottom right image).
Illustrator also have similar issues - see screenshot below - less or more visible depending the type of preview you are using: CPU preview gives better results than GPU preview - you can switch between them pressing ⌘ (cmd) + E). In the example below i created two objects - a rectangle and an ellipse - and performed a Divide operation, then placed a black rectangle below so you can see the issue more clearly.
You may also see hair thin lines which are related with the way Affinity renders the objects on screen/zoom level (different issue). If you check pixel aligned objects at 100%, 200% 400% and so on (x2), those hairlines will never appear. With arbitrary zoom levels this becomes a rendering issue (when the edges of two objects overlap). However in this case the file exports correctly, the lines only appear on screen. MattP is the most suited person to give you an explanation about why this happens and what can be done.
