Your own server:

I feel the cost of hosting your own server can be an issue, since I've done it. 

Cost and upgrades, maintenance and upkeep, as well a streaming and software updates and of course downtime, is all in your hands, as well as internet connection issues. On top of also running your site.

Paid host server:

They are responsible, for upgrades, repairs and configuration. They also carry any overhead involved with server and internet issues. They usually are running the latest and best equipment and software, that you aren't responsible for doing.Which leaves more room for you to manage your site.