(I don't use it myself. I know people who do.)
the practical benefit is that your ISP will change your IP address (e.g., Telekom Germany, every few days, I think) but the DNS name stays the same.
the security implication is that you now have one more place where your actual IP address (of an open port) is known, so it increases the DOS risk.
if the server is just used for streaming, then there can't be much damage (at worst, you re-install it)? well, 1. it's an entry point to your internal network, so other machines (and data) are at risk as well 2. an attacker could hack your server (silently) to do something illegal, and you might only notice when the police come knocking.
If you rent a server instead, you can avoid 1 (mostly - but you still have to connect to your server in some way, and that's the point where you are vulnerable) but you still have 2 (the hosting provider knows you).
I do streaming from work, as part of my job (teaching), and our IT staff monitors the network.