If you XFinity subscribers reading this don't already know, you could already get Showtime content on your Kindle Fire via the XFINITY TV Player app. That app also gives you access to almost all the premium channel content that you subscribe to. So, what's the difference between the two apps? For starters, I'm sorry to say that neither will let you export video to your TV. Unlike HBO GO and Max Go, both XFinity and Showtime decided to be a sticks in the mud and block this option. Personally, I found the navigation in the XFinity app to be easier and more intuitive than Showtime Anytime. Generally speaking, the amount of content is almost the same. Something along the lines of 137 vs 139 movies, and 17 vs 15 shows. That's in favor of the Showtime app, by the way. A brief look says that the difference is in what each offers for the Showtime network's "After Hours" section. Playback quality is equal between the two apps, and both apps have the same amount of stability. Really, unless the minimal extra "After hours" content that you can't live without, I don't really see any reason to pick Showtime Anytime over XFinity TV Player. If they start allowing export to TV, then that would be something at least.
*Note: Apps can get updates and change over time, for better or worse. Please note the date of this review, as it may contain info that is no longer accurate.*