Making the Home and About pages different should be pretty easy.
So, to start with ... it isn't that they are just reading the same ... those ARE the same page. So all you need, is to create a new, different page to be your about page, and then to update your menus to point to that page from the "About" button.
I wouldn't worry about starting from scratch, either ... if you can't rebuild it better in half the time you took to build it up on the first place, you probably should be taking better notes along the way! But, it is always worth a little time trying to undo mistakes you've made along the way, it helps you understand the way things work better.
A lot depends on your theme. Some themes have things in them you don't want or need, and some create things you start to rely on ... even if they aren't WordPress standards.
To fix your Home-About issue you would need to figure out a couple of things, first (which of those pages actually exist, and how you have your menus set up) - so to start, from the dashboard go to Pages -> All Pages, and check to make sure you don't have a separate Home and About page already. With one theme for one of my sites, the Wootique theme creates a Home page more or less automatically, so it doesn't even appear in my list of pages. Still, I know it is there ... and the theme options let me populate it with blog posts, featured products and a list of products. You should figure out what pages are created by your theme or plugins, but if you don't know them all right away, just make notes about them as you go.
Assuming you do not have an About page already, click Appearance -> Menus, note which menu area you are using, see if there is a custom menu being used in that area already. If there is a custom menu being used, edit it and get rid of the About page in that menus. If there is not a custom menu being used, create one that suits you but leave the About link out of it for the moment.
Then go to Pages -> Add New and create a new fresh, different "About" page. Put some copy on it or an image so you can distinguish it from your old one.
Then go back and re-edit the custom menu to include your new About page.