Facebook share and Facebook like are different but effective ways to share websites content on Facebook. Like and Share functionality seem to be similar, but they are used for different purposes.
This is the original way to share links of your website on your Facebook profile. With this functionality you are able to add a comment about the link. In your Facebook profile, a comment, title, description and thumbnail of your link is displayed.
However, Facebook is encouraging developers to use Facebook Like instead of Facebook Share. This is the official link to the Facebook Share documentation
With Facebook like you share links of your website on your Facebook profile, and it also appears on your user’s friends’ News Feed. In addition to that, if your website represents a real entity (an actor, a company, a restaurant, etc) you can configure your website so it works like a Facebook page.
This is how Facebook like looks on your site:
This is how it looks on Facebook:
This is the link to the official documentation about Facebook Like.
In this link you can learn how to configure your website so it behaves like a Facebook Page
In conclusion, Like functionality is the current way to share content on Facebook. Although Facebook Share allows you to publish more content of your page on your wall, Facebook Like allows you to relate your website with your Facebook page, and reach more audience.