First - let's clear up the difference between shared and dedicated hosting. Shared hosting is usually the cheaper option, and one preferred by many small businesses and personal sites. With shared hosting several (sometimes hundreds) of sites are hosted on the same server. If that server goes down or a site on the server experiences high traffic volumes, then all of those sites could go offline or run slower.
However, larger businesses with more site traffic, usually require dedicated hosting. That means that your site is the only site stored on that web server. But how does a dedicated server help?
Handle more traffic
In theory, dedicated hosting means that you can handle more traffic. The overall load on the server is based solely on your site visitors, not the visitor total from a group of websites. Dedicated space usually allows for more bandwidth - ideal for sites with lots of content and pages. It also means that your site is more reliable for your customers, with less stalls or downtime from traffic spikes.
Security
Shared hosting puts you slightly at the mercy of other people. If their site gets hacked or they somehow upload some malicious code to the server, that affects everyone on that server (including you). With a dedicated server, only you have access to the server, theoretically improving the security of your site. You usually also have the flexibility to add more security software such as a firewall.
Of course, if you're running an ecommerce site, or collect customer data, this added security becomes crucially important.
All dedicated hosting is not created equal
There are plenty of hosts out there. If you find a good website development company they may arrange hosting on your behalf.
The key is to check what's included in your package. If possible, ask one of your tech team (particularly a system administrator) to look at the contract details.
For example, is your site regularly backed up for free? What about recovery and remote reboots? Some hosts will charge extra for these services - and this can quickly add up to a lot of money. Be sure to check what's included in the monthly fee.
As with anything to do with website design or web hosting, cheap is usually not your best option - and may end up being more expensive in the long run.