The big difference is in how Medium hosts it's content versus how Wordpress.com hosts content. With Wordpress.com users get a subdomain and all the search engines treat that subdomain as a totally new domain.

With Medium you post an article and it's hosted on the TLD Medium.com followed by the user and an article name. This is an important distinction between the platforms since sites and users treat Medium.com with an increased level of importance.

My startup SHAMELESS PLUG ALERT http://www.joinjune.com has been using Medium posts with incredible results. We have had about 300k views on the content in the last few weeks. We posted the same content to our blog at first http://blog.joinjune.com and barely had any traffic at all.

Granted, our blog is really ugly (Thanks to Hubspot's unnecessarily complex template system) but also in part due to the fact that sites like Hacker News treat Medium more fairly and people feel comfortable sharing articles on there.

We have even had of our articles reblogged by the New York Observer, MatterMark and translated into foreign languages because it was so successful on Medium: https://medium.com/@ayasin/the-one-method-to-eliminate-bad-t...

I'm a fan of Medium. I tell everyone to use it instead of their own blog based on our successes with it.