How does screen resolution affect what you see in terms of print?

First, the most accurate way to view anything (Photoshop, Lightroom Classic etc) is at 100% (or 1:1). Zooming out subsamples the preview and can produce a less accurate preview of the image data. 

As for DPI vs PPI etc. There is zero difference in a document that is 1000x1000 pixels (as an example) at 72 dpi (PPI) or 720 dpi (PPI) or any such value. All are 1000x1000 pixels and the dpi/ppi is simply a metadata tag. 
This very, very old primer on resolution still seems necessary to post, this may help in understanding the differences:
http://digitaldog.net/files/Resolution.pdf

The screen resolution is used to calculate how big the document will appear when using View > Print size. The aim is to enter the value so that the rulers at Print Size look physically correct (20cm = 20 cm on screen). There is also an Actual Size which picks up the screen resolution automatically, if the screen is passing it through the operating system correctly.

As for ppi versus dpi. Images are made of pixels, so ppi is the correct term for printing - literally how many pixels will be printed in 1 inch. 

dpi is often interchanged for ppi, but is incorrect to do so, as it describes the dots that will make up each pixel. So set ppi to the value you require for printing and that is it.

You will see some inkjet printer documentation saying the printer can print at 2800 dpi. That is correct, but they cannot print at 2800 ppi (remember dpi is dots not image pixels). A max of 720ppi on those printers is common sometimes 300 or 360ppi

Dave

Your screen resolution only affects how the image is displayed on screen, not how it prints. This is in reference to the display control panel on windows, and the finder preference on mac.

There are settings that affect both like effect >> Document Raster Effects Setting. Illustrator for the most part is a vector program and vector items print at max resolution of the printer. You can place bitmapped images and create bitmapped effects like gaussian blur, those print at the resolution they are at.