What do you like best?
Deploying a web application by *pushing* its associated git repository master branch is very convenient and feels so natural. I knew Heroku users could do this and always envied them for this reason alone! Using Heroku for all my projects have been a no-go for me because of the cost. Dokku offers a similar platform for free! I can install it on a cloud platform such as DigitalOcean, which is very affordable for freelancers or small development teams.
Thanks to Docker, which Dokku is based on, your app(s) lived inside isolated *containers* which give you the portability of a virtual machine, but lighter and with less overhead.
Another great thing with Dokku is the plugin system. You want Postgresql database support? Just install a plugin using the Dokku built-in command: dokku plugins:install plugin-git-url.
Dokku 0.5+ also allows you to run pre-deploy and post-deploy commands using an app.json manifest (as Heroku users can do). This mean, for example, that you can run database migrations automatically on deploy, which you couldn't do before 0.5.
I also like the facts that Dokku uses the same buildpacks as Heroku.
What do you dislike?
Honestly, there is nothing I dislike about Dokku at the moment. Before version 0.5, there was the fact that I had to run my Rails database migrations manually once the application is deployed, which was quite inconvenient. But now that it is fixed, I'm happy.
Recommendations to others considering the product
If you go the Dokku route, keep in mind that your server will have to be administered using mostly Dokku commands... not standard Linux/Unix commands. For example, to check the log files of your application, you don't "cd" to the app directory and check the logs there. Well, you *could*, but you would have a hard time trying to find where your running application is located exactly. Instead, if you want to check your application logs, you use a command like this "Dokku logs myapp". Same thing applies for configuring vhosts, environment variables, etc.
Also, if you don't agree with the "twelve factor app" mindset (http://12factor.net/), you might be disappointed.
What business problems are you solving with the product? What benefits have you realized?
With Dokku, I am way more productive in my workflow. This piece of software is perfect to run on a cloud platform such as DigitalOcean. I can isolate each project/client on its own Dokku-ready "droplet" instead of having them all configured on one or more dedicated servers.
I personally use Dokku for Ruby On Rails applications and let me tell you it is a pretty good match. I have written a tutorial on my blog to help people getting start with Rails and Dokku (http://www.rubyfleebie.com/how-to-use-dokku-on-digitalocean-and-deploy-rails-applications-like-a-pro/).