2. Ghost actually lets you have a brand
On Medium you're locked into one design with some basic settings. If Medium decide to change how your blog looks and works (which they regularly do), then you have no say in the matter. Your Medium blog looks like every other Medium blog, and has Medium links and logos all over it.
With Ghost: You choose your design, you choose your branding, and you choose your links. Each Ghost blog is unique, and reflects the personality of its creator.

On top of that, downtime is limited by each blog being separate. If Medium goes down: All Medium blogs are down. If a Ghost blog goes down, none of the others are affected. It's far more stable and reliable to host your publication in a place where you know it's safe from attacks.
3. With Ghost, you control your content
When you blog on Medium, you give up control of your content and ownership of your traffic. While it can be satisfying to see your post being widely read thanks to Medium's network effect, it's Medium themselves who ultimately benefit from your work. With Ghost, all of your content, traffic and branding is your own - and setting up a new Ghost blog is just as easy:

Because Ghost offers an official hosted service, you can have a new blog up and running in the space of about 3 clicks. Powerful automation saves you a lot of time getting a custom blog deployed.
4. Build a business around your blog
With Ghost there are no limitations on what you can and can't put on your blog. If you want to use display advertising, that's fine. If you want to collect email addresses as leads to drive to your business, that's fine too. If you want to create a platform around your blog to start selling your book, that's A-OK.
If you're looking to graduate from the simple social-network style blogging of Medium to more serious publishing as a career: Ghost gives you a lot more flexibility.
5. Ghost has SEO & social built-in
What’s the point in publishing if nobody reads your content?
Ghost has the best support for search engine optimisation and social sharing features of any blogging platform in the world. You don’t need any extra plugins or extensions, and you don’t need to write any extra code. It just works.

There are built-in XML sitemaps, Google AMP pages, canonical tags, optimised URLs, microformats, Facebook Open Graph tags, support for Twitter cards and clean semantic markup. All of this is done for you automatically, with no plugins needed.
"We’re using Ghost’s Pro service to host the blog for us. Works really well. Great writing experience compared to WordPress. WordPress is an incredibly powerful framework but because it supports the needs of so many websites and demands, it has grown into a sizable and cluttered tool."
Switched from WordPress to Ghost