Separate projects for the web api and the web interface will help split things up, but it does cause duplications. We went that way recently and it works well, but it caused a few problems.
Arguments for having a single project :
Since we don't have a domain name yet, we have our API on the 8080 port. We could use a directory binding to make the API accessible from a sub-directory of the web interface but we were worried about production only bugs about absolute path resolution.
Many settings are shared between the two projects so we have to copy them in both web.config files.
Arguments for having multiple projects :
They are easier to upgrade since they can have different dependencies and they can be built totally independently. For example, our API project uses a few more recent versions of some dependencies.
It forces you to extract all of your business logic into a separate library and makes it easier to think about both projects as separate sub-systems.
It is easier to setup the web interface to a separate machine if the load is too much. This is a concern for us, but that may not be your case.
If I had to make this decision again, I probably wouldn't bother with separate projects unless the system was extremely complex and I needed the additional structure. An argument can be made for both options, but I think the deployment headache it brings is not worth it.