The primary difference between the "Simplified payment page" and "Side-by-side payment section" is the positioning of the page content's payment forms.
The "Simplified payment page" offers the simplest client experience by only showing the payment forms, PDF download and amount owed on top of the page. To see the full invoice details itself, the user must scroll further down the page. The benefit of this is that the high-level details and payment forms are shown with less distraction.
The "Side-by-side payment section" shows the full invoice and payment forms side by side all at once. The benefit of this is that your client can see everything easily on the page without scrolling. The trade-off is that the payment forms and PDF download button are displayed in a less prominent way on the page.
At the end of the day, it is all preference. You may want the payment experience to be as simple as possible for your clients, or you may want them to immediately see the full invoice details instead.
Changing Client Layout Settings
To change your client layout settings, you just need to visit your Invoice Settings.