In short
- At home it is your upload speed that counts, not download, and you have to forward ports on your router.
- Your PC has to stay on and your IP is exposed to attacks.
- Renting has a clear price and does not depend on your machine or your connection.
Your connection: upload, not download
A server sends data to every player, so what matters is your line's upload speed, which on a home connection is usually far lower than the download speed. A few friends may be fine; a community will outgrow it and everyone will feel the lag.
Opening ports and your home IP
For outside players to join you have to forward ports on your router. Some providers share one address between several customers (CGNAT) and do not allow it.
Also, anyone who connects to your server learns your home IP. In games where attacks are common, that is a real risk to your everyday connection.
Your PC has to stay on
If the server lives on your computer, it only exists while the machine is on and the game is running. Playing yourself while hosting uses memory and processor, and a reboot or a power cut takes everyone's session down.
Protection against attacks
A serious rental service filters malicious traffic before it reaches the server. At home you have no such layer: an attack can knock your whole household offline.
Real cost
Hosting at home looks free, but count the electricity, the wear on your hardware and your time to maintain, update and fix it. A rented server has a clear price and goes away when you stop paying.
So which one?
If you play occasionally with two or three friends and your connection allows it, you can start at home. As soon as you want the server always available, with more players, or you need to protect your connection, renting is usually the better option.