There is a common belief that redesigning an existing website should be much easier than starting from scratch and creating a completely new website. What any designer who ever undertakes a redesign eventually learns is that it can be much harder to redesign than to design from scratch. Yes, you have some raw materials to work with and the website has an established vibe. That is exactly what turns out to make the job difficult!
When you start designing a fresh website you are coming up with the vibe it gives out to visitors and the flavour of each page as you go. You can create whatever you want and keep changing things until it is just perfect. When you redesign an existing website you don't have as much freedom because you don't want to drive away visitors who have already come to love the site. You have to stick with the existing flavor and vibe while creating something fresher.
This isn't always an easy task to carry off. If you do a complete overhaul and give the site an amazing new look that doesn't look much like the old site, you risk losing visitors who no longer feel that connection with the site that they had previously established. Some may even be confused and think they have come to the wrong site or they may assume the site has been taken over by someone else. That often happens when you change the content along with the basic design.
The easy answer is to keep certain elements of the old site but make them a fluid part of the new site. You may use the same basic design but with a different color combination or the entire design may be different with the use of some old graphics and smaller design features.
It is also important to keep the logo or branding of the site the same or as close to the same as possible. This way, visitors will at least notice the logo and know that they have come to the right place. If you go with a new logo, name, or slogan, then consider it a complete restart for the website because many old visitors will treat it like a new site.
You don't have so many problems when you go with a completely fresh website designed from scratch. There are no existing site fans to worry about and there is no established logo or name that you have to work into the new site. You are not confined by any style elements of an older site. You have complete control to just design and create what your client wants (or what you want for your own sites).
Designers typically are not given a choice between redesigning an existing site or starting from scratch. Clients either want to start a new website or they are interested in redesigning an older site that they no longer like or which has grown too cluttered over the years. When presented with a redesign job it is important not to assume it will be easier, or just as easy, as designing a new site. There are different problems that present themselves and the process of design is a bit different.
You should communicate the issues of redesigning with a client and figure out how much of a change they really want to undergo.