Benefits: your data will be yours, no matter what apps or companies shut down. You can create as many accounts as you wish and do whatever you want among them. Bridges exist that allow you to use different chat apps all through your own XMPP server. XMPP clients are often open source and have many features.

Downsides: barely anyone uses XMPP anymore. Setting up and maintaining a server requires some work, especially with STUN/TURN. You’re responsible to prevent downtime, there’s no million or billion dollar company keeping the servers going. You need to figure out backups (both client side and server side). Playing XEP bingo to find a client that supports what you want can be a pain.

If you’re interested in bridging other chats, I’d recommend looking at Matrix instead; it’s receiving more community attention so there are more up to date bridges.

If all of your chats are end to end encrypted and you trust the server of your choice to stay online for the coming years, there shouldn’t be any downside to sticking with someone else’s server. If you have unencrypted chats (with bots you run, for example), running your own server would be more secure.