Performance
You are not sharing the system resources such as processor
power, memory and disk space with other sites on the
same server.
Greater Reliability
You do not run the risk of the server being slowed
down, swamped with traffic, or crashed by another site
on a shared server.
Security
As nobody else has access to the server you can be
sure of having a much greater degree of security and
you have the authority to install additional security
software on your dedicated server.
Custom Configuration
On shared hosting, you are not allow (in most instances)
to install any special software or configuration that
your script needs. In most instances, dedicated server
will be your only options.