In many cases, sites sending these notifications with the proper microformats mark up means that you can get some really beautiful replies to show up in your comments section (esp. in relation to how the old linkbacks/trackbacks looked). Webmention also has some structure as well as potential extensions to prevent the spam that the prior implementations encouraged.
If you reply to my syndicated copy of this post on Twitter, I’m also using the free service Brid.gy to have Twitter send these notifications to my personal website, so I’ll see your reply on my original post without actually needing to visit Twitter directly. This means that not only can I do threaded replies between my site and another WordPress site (or any other site that supports Webmention), but I can do threaded conversations between my site and Twitter.
Now if you want to take this the next few logical steps, add Micropub support to your website, and start using a social reader like Indigenous. That will let you write replies to content in your reader that will automatically post those repsonses/replies to your website, but then your site can ping the site you were responding to! The specifications allow a true social media experience between websites running different software on different URLs. Some documentation for the WordPress side of things: https://indieweb.org/Getting_Started_on_WordPress
The more sites that support these specifications, the richer the ecosystem becomes.