In Photoshop (and some other drawing programs), when setting the canvas size in mm or inches for printing intent, the canvas appears larger on the screen at a higher resolution. A4 @ 300 ppi appears larger than A4 @ 72 ppi, and the exact same brush size appears smaller on the former than on the latter as a result. Why is that? I guess because the pixel size is fixed, the mm or inch metrics have to change accordingly to the resolution, but does that change occur only on the screen, or does it also affect the dimension of the print in any way?
It's because . . .
A4 at 72ppi is 595px x 842px
A4 at 300ppi is 2480px x 3508px
The number of pixels in the width/height is determined by number of inches multiplied by the PPI. A4 is 8.27" x 11.69"
So basically, an A4 at 72ppi has a smaller canvas size in pixel dimensions than one at 300ppi. So if your brush is a specific size, it's going to look bigger on a smaller document.
Edit, to clarify:
All that matters in raster images ultimately is the number of pixels. You can think of a higher PPI as like a conversion factor used to cram more pixels into a smaller space - i.e. more pixels per inch. The PPI alone doesn't actually affect the number of pixels. It only does so when used in combination with a physical size unit, measured in inches, when used to work out the size of a document in pixels, and the size it will be output at in print.
For images that are to be displayed on a computer screen, the PPI is irrelevant. Images are generally displayed at the native screen resolution of the device being used, assuming no zooming/scaling is taking place.
On 300 ppi a 72 pixel brush will take 1/4 of an inch. While in 72 ppi it will take one inch. So Photoshop show that difference accordingly.
The length and width of your artwork is fixed. It's either a4 or Letter or else. The amount of pixels change in the file itself. While your screen resolution is set. So an A4 300 ppi will have 2478 pixel in width. While your screen may have 1440. Zooming out to see whole page require program to recalculate what you see. While 72ppi would take 595 pixels. One pixel in artwork using only pixel of your screen can be done without zooming in or out.
SO: It would only affect final size of print IF your printer could choose paper size depending on PPI with stiff set of pixel per inch. SO it would do exactly opposite of what resolution is used for.