Different people have different needs. There are people for whom a shared server is sufficient and there are others who need a dedicated server for hosting their website(s). Within each of these categories too you have several options. So you have several web hosting plans for shared servers and you also have various kinds of dedicated servers (that come with different capabilities). It’s all dependent on your need and budget. The main difference between these two kinds of web hosting is in terms of server space and the transfer allowance.
A number of people start with hosting their website(s) on shared servers and as the hits to their website increase or the size of their website increases (on account of inclusion of more graphical features and video/ audio clips), they feel the need of getting a dedicated server. A dedicated server dedicates its full capabilities to the website(s) that it hosts. Hence, they are able to increase the performance of the website(s). However, it is important to note that the cost of hiring a dedicated server is much more than the cost of getting space on a shared server. So, the decision to move from a shared server to a dedicated one should be a well thought one.
You might be planning to upgrade your website e.g. you might decide on adding more graphics to your website or you might be putting up some video or audio demos on your website. In all such cases you will need more server space and transfer allowance. However, that doesn’t mean that dedicated web hosting server is the only option for you. Before you move to a dedicated web hosting server, you must evaluate whether any of the other web hosting plans (shared server ones) will satisfy your need. You will need to do a proper cost-benefit analysis to determine if it’s time to move to dedicated web hosting server. Minor difference in costs can be ignored in favor of additional convenience that visitors to your website will get. However, a large differential in cost calls for strong reasoning behind going for a dedicated web hosting server. Also consider your future plans. If you are planning to add a lot more content in the near future, you might find it easier (and more time-effective, effort-effective) to move straight to a dedicated server instead of first moving to a higher plan on shared server web hosting and then to a dedicated server web hosting.