Common input boxes to register and login with separate buttons

While working on the registration/login page for my mobile app, I came up with the idea of combining the two into one. On clicking register, users will be taken to a new page to fill rest of the details.

I wish to keep user onboarding as simple as possible. For example using the conventional approach, users see a login page first. Those who wish to register have to click a Sign up instead button which opens up a new page.

By using my approach users will be saved from an extra click and a page load. But what are the pros and cons here?