Your domain name is your new business address but without a dedicated server, you cannot serve your customers with your best capability. If the domain name is your new business address, then a dedicated server is your place of doing business, a small place can only handle a limited amount of traffic at a time whereas online possibilities are endless. The common problem with all small hosting options is that when your business goes up your website goes down.
Whereas on a dedicated server, the server is configured exclusively for you where you can host your business website as well as sites for your family members, your smartphone app, and your online performance will not get affected by your success.
Apart from this, with super admin access, you can better control your online business by configuring your server to best fitting your web/app requirements. You can also assign a programmer or manager to your server without giving them the root access, this allows you more security. On a dedicated server, your online business can continue to scale newer heights as it can very well handle sudden raises in traffic.