Now that Web Author supports most of the functionality of desktop Author (and surpasses it in some areas), the most common reason we've been given by customers for not using it has been that the performance did not match the performance of desktop Author. We've recently done some work in this area for one particular case, model load speeds.
We found a particular model (consisting of 75 diagrams!) that took 25 seconds to load in desktop, but 3 minutes to load in Web Author. After our optimisation work, Web Author now loads the model in 20 seconds. Obviously this is only a single operation out of many that can be performed (although the changes we have made should improve performance in other areas), so do let us know if you find any other areas that you feel are slow. Tell us the browser you're using, the model you are working on and the operation that is slow.
Other ways to improve performance can be as simple as upgrading your browser (we realise this isn't always an easy thing to do depending on your IT department!). Browser vendors are in a constant battle to top the performance list, so the latest version of a browser will almost invariably be faster than its predecessors. So if you use Internet Explorer, IE11 is likely to provide the best experience. If you use Windows 10, Microsoft Edge is likely to be better still. In our experience Google Chrome provides the best overall performance