The key to good print quality is the creation of an ICC profile for your printer/paper pair.
Photoshop Elements, with a few exceptions, allows you to print as in Photoshop, i.e. using custom ICC profiles. This is a significant guarantee!
The printing options must be exactly the same as when you printed the test patterns used to create your ICC profile.
If you want the highest printing fineness, choose 2880 dpi (for Epson or Canon printers): this allows you to have more beautiful and progressive gradients in the sky, in particular, especially visible at close range, so on small format prints because you get closer to them. It's subtle but, personally, I see the difference and I clearly prefer it.
The dpi of the printer have nothing to do with the dpi (ppi) of the image. If you want to print a photo at 240 dpi, the gradients will be more beautiful if you print it at 2880 dpi (printer) than at 1440 and even more so at 720 dpi. They will also look better at 240 dpi (image) than at 120 dpi.
Beyond 240 dpi for an inkjet print, I don't see any difference but from 200 dpi I start to perceive it if I get closer to the print run. 300 dpi is only used for offset printing at the printer's but does not contribute anything to inkjet printing.