In both cases you are providing an assessment of the market and of the competitive position of the company in that market. While there are potential differences in the type of content and scope (read my last paragraph), the major difference is in the process. 

The biggest differences are:

• Access and interaction with the company's management
• Data sources (more internal vs. more external)
• Expert use
• Schedule flexibility

CDD: you have a limited number of touch points with the management and they won't have access to whatever you are writing in your report. You will have lower access to their data (other than the company's financials), and will resort to a large number of expert interviews to understand the market and competition. Usually the project is shorter and with a fixed deadline (i.e., no ability to extend).

VDD, you have significant access to the management team. You can have regular calls to understand their business and they may be the starting point to understand the market and competition, as well as company's positioning and strategy. You will resort to experts as well, of course, but mostly to confirm that information. You may also have access to their clients in calls arranged by them (this can also happen in a CDD, but more frequent here).

Since they are your client, you will always present your findings before you issue the report and as such provide them with the opportunity to address any red flag or relevant risk you identify. You never do this in a CDD. So this means they have a better opportunity to positively influence your conclusions. The project usually takes a bit longer (i.e., 1 extra week) because they are gathering information on their side for the first time and it is also more intensive on their side. Moreover, they need time to read thoroughly your report and gather additional information if they think that is necessary in order to help presenting their case in a better light. If necessary, they can slightly extend the scheduling.

In both cases you are independent. But in a VDD, the company has a stronger ability to make their case. Of course, in a CDD you will usually start by looking into the VDD and then challenge it.

Having said this, in some instances the CDD is actually testing a very specific investment thesis (e.g. can they be successful if they enter XYZ market? can you do a deep dive on a specific narrow topic?) I.e., in some cases you have multiple CDDs on a company, and the first one is more or less what I described above (overall market and company) and there's an overlap between the CDD and VDD, but then in the subsequent DDs you start working on topics that weren't yet addressed by any DD.

Hope this helps!
Pedro