

Not really, you’re ideally paying for a server that you have complete control of. The differences are mostly just fundamental limitations.
Example: if you’re hosting off site, you will always be connecting remotely, so your access depends on a network connection. If you’re hosting at home then your stuff is still accessible when your internet goes down
You can do a lot better by buying your own modem and router, but that can be expensive. The thing you’re doing right now is a good idea if you don’t want to spend a lot of money, whine at your internet provider and get them to send you a better router.