There's no more editing a checkout.liquid template directly. Shopify phased that out in favor of checkout extensibility: a set of defined slots in the checkout flow, above the line items, below the shipping method, on the thank-you page, and others, where an app-built UI extension can add content. You don't touch raw checkout code; you build or install an extension that renders in one of those slots.
Shopify Functions is the piece that handles logic rather than visuals. A function written against Shopify's Functions API can change which discounts apply, which shipping rates show up, which payment or delivery methods are available, all computed server-side based on rules you define. It's how merchants get custom discount logic or shipping rules that Shopify's standard settings don't offer out of the box.
The catch is plan tier. Checkout UI extensions are broadly available, but the deepest customization, full checkout branding beyond colors and logos, custom validation rules, and several Functions use cases, is reserved for Shopify Plus. If you're on a standard plan and hit a wall, that's usually why.