Authentication screen is native UI - credentials are sent to your server using REST
Loading the UI in a Webview, and programmatically get the cookie in your mobile app upon a specific event (e.g. when redirecting to the success page)
Native UI forces you to duplicate some code (web + api authentication endpoints) whereas the webview might not be so good on a UI/UX perspective (at least you’ll need to work out the login view to make it fit mobile screens and UX). Both solutions are ok as long as you own all of apps, i.e. no third-party (Openapish) accessing your data.
How do you intend to access data from your mobile app? Loading web pages and parsing them? Or on dedicated APIs?
Should you have app on different domains (cookies are not shared between domains) or you want to allow access to third-parties, you might consider using OAuth2 & OpenID Connect.