RStudio Connect is on-premises software you install and manage within your organization. shinyapps.io is a multi-tenant Software as a Service (SaaS). You can find a feature comparison here: http://www.rstudio.com/products/shinyapps/.
In general, if you are comparing RStudio Connect and shinyapps.io, you'll want to ask yourself these questions:
1. Are you okay with your application being outside your firewall?
2. Are you okay with the data that the application is pulling from being accessible to our cloud?
(You have to open up a hole in your firewall if the data is behind the firewall today.)
3. Are you okay with your end client creating a user account on shinyapps.io (if you are looking to use authentication).
4. Are you okay with a shared computation platform for your analyses? (for example, we don’t have any SLAs today on performance)
If the answer is no to any of these questions, you may want to go with RStudio Connect.