I'm wondering, What are the differences between customers "liking" a webpage and customers "discussing" it. For instance let us say I've got a news website with a lot of articles. Should i've got a "Like" button around the home page along with a "Share" button on every individual article? Or opposite holds true?

Why Facebook suggests while using Like button (beware, sarcasm inside) :

Why you should not make use of the Like one anyway :

Facebook does not recommend while using Share button, rather while using Like Button. This is exactly what Facebook states concerning the Like Button.

So On button allows a person share your quite happy with buddies on Facebook. When the consumer clicks so on button on your website, a tale seems within the user's friends' News Feed having a link back aimed at your website.

More information and tutorial can be found in the link above.

Hope this can help.

an essential differentiator is performance

a share-link doesn't not make any request to facebook (or other site) unless of course clicked on, as the like-widget makes demands to facebook (hence the privacy problem Chouchenos rightfully mentions).

this indeed implies that a webpage using the share-link will load faster compared to same page using the facebook like-widget.