The difference isn't clear to you yet because you've published only one post.
When you click Preview while editing a post, you view a single complete post. When you view your "Blog" page, you view an archive page.
The two views aren't the same: the single post view displays the entire content of the post plus its comments section; an archive view displays a series of posts (in reverse chronological order), it doesn't display the comment section of each post, and it can display excerpts instead of full posts.
When you're on an archive page, you move to a single post by clicking the title of the post, or its comment link, or (if the page displays excerpts) its "Continue Reading" button.
Displaying Likes and Shares is optional. Enabling them on the single post view and enabling them on archive pages is two separate options (set in Settings > Sharing).
If you prefer full posts on your Blog page, follow Jennifer's suggestion.
If you prefer excerpts on your Blog page, don't change the settings; instead, edit the post and insert the more tag to truncate the post, so that visitors will have to click "Continue Reading" to see the entire post (including its Likes and Shares).