Domain names and web hosting seem to go together like peanut butter and jelly. You buy a domain name and connect it to your web hosting. It seems natural to have the two together. But I generally advocate against keeping your domain name registered with your web hosting company.
The main reason I advocate for keeping your domain name(s) separate from web hosting is because things are far more likely to go wrong with your web hosting before your domain name. Domain registrars are at least partially regulated, by ICANN. There is only one documented major registrar failure, RegisterFly, and stricter oversight has been put in place because of it. Domain registrars aren’t perfect, but for the most part they serve one purpose: managing a domain name for a small fee. The majority of them are adequate at this function. Registrars have also standardized the process of transferring domain names between registrars. So switching registrars isn’t terribly complicated (get transfer code, provide to new registrar, accept transfer). The registrar also has no influence on your domain name*.
Web hosts have full control of what your website can do. Almost all of them have restrictions in their terms of service (especially shared hosting) which limit what you can and can’t do. If they have your domain name too, you can’t simply change web hosts in the event of a problem or dispute. Your domain can be held hostage or as a bargaining chip to make sure you renew or pay them.
There is potentially a secondary problem with a web host registering a domain name for you. The ownership of the domain name comes into question. Do they register it in your name making you the legal owner or their company’s name, making them the de facto owner.
The common arguments in favor of keeping your domain name with your web host are simplicity and cost. I can’t argue against the simplicity argument. If having two accounts with different companies is too difficult for some reason, there is no counter argument. The cost argument shouldn’t come into play when a domain name is around $10/year. It’s a fairly low cost item relative to the amount of pain it can cause if something goes wrong.
In conclusion, unless you have some exceptional circumstance preventing you from keeping your domain separate from your web hosting, the two simply don’t belong together at one company.
* Some registrars do have policies restricting the type of name, the content put on the domain and the activities a domain can be used for. These generally include spamming and illegal materials.