What can push notifications actually do for a Shopify store?

They reach a customer on their home screen without paying for the reach, which no other channel does. That is the app's single biggest commercial advantage over a mobile site, and the reason an app is worth building at all.

Email is filtered and paid ads are rented. A notification is the only route to somebody's attention that you own outright once they have installed the app.

It is also the easiest thing to ruin. Three notifications a week and the app is uninstalled; one that matters and it is opened.