I think the reason is not security nor startup time of the app. Let's understand what's behind the scene before we find out the root cause.
Java Control Panel has settings that allow users to use default browser's proxy settings or override settings. In other words, infrastructure teams are able to customize the Windows or OS installation images to have JVM pre-installed with enterprise proxy settings. So I believe this is not an issue at all.
Java Web Start actually caches all apps with customizable settings in Java Control Panel. Once the app is cached, the app is "installed" just like other apps. Although first time execution may be slow, the second time will be fast due to JVM's smart memory allocation technique. So start up time could be an issue but a lot of web sites (even enterprise internal) are now migrated to portal. A web portal normally contains lots of unused libraries for development purposes due to the fact that portal itself does not anticipate what kinds of portlets are built and deployed on specific page. Therefore, downloading a single portal page could consume up to MBs and complete a page in more than 5 seconds; this is only one page and caching helps up to 30% but there are still lots of HTML/Javascript/CSS components are required to download every time. With this, I am sure Java Web Start is an advantage here.
Java Web Start does not download again if it is cached as long as the server copy is NOT upgraded. Therefore, if, e.g. a project management software like MS Project, is completed using SmartClient (similar to JWS), the information exchange between the client and server would be purely data without presentation like browser's full page refresh. Even with the help of Ajax, it doesn't eliminate full page download entirely. Besides, a lot of companies consider Ajax to be immature and unsecured still. That is why Ajax is a hot topic in the circles of developers but not within enterprise software yet. With that in mind, JWS apps definitely have more advantages such as JWS apps are deployed and executed in sandboxes, signed, and much more interactive GUI.
Other advantages are faster development (easier to debug in code and performance), responsive user interface (does not require Comet Servers to provide PUSH functionality), and executing faster (for sure because client computers renders GUI without translation like HTML/Javascript/CSS, and less data processing).
After all these, I haven't touched the question yet, why JWS is not so famous?
My opinion is that it is the same as Brian Knoblauch's comment, it's without awareness.
IT folks are too attracted by the hype of Web Technologies, Ajax PUSH, GWT, and all those buzz words make them bias towards the fun of using different technologies or to resolve technical challenges instead of what's really working for the clients.
Take a look at Citrix. I think Citrix is actually a great idea. Citrix allows you to build your own app farms behind the scene. There are tons of upgrade and implementation strategies you can go for without impact to client experience. Citrix deployment is extremely easy, stable and secure. Enterprises are still using it. However, I think JWS is even better than Citrix. The idea of JWS is to run apps on client machines instead of hosting tons of server farms where client machines are capable of running these apps themselves. This saves a company a lot of money!!! With JWS, development team can still build business logic and data on server side. However, without the web processing unit and let the client computers do the rendering process, it greatly reduces the amount of network consumption and server processing power.
Another example of why JWS is an amazing idea is Blackberry MDS. Blackberry apps are actually Java apps translated from Javascript. With BB's MDS studio, you use the GUI tool to build BB app GUI, coding GUI logic in Javascript. Then apps are then translated and deployed on BES server. Then BES server will distribute these apps to BB. On each BB, it runs a thin Java App with GUI rendering and networking capability only. Whenever the app requires data, it communicates with BES then through web services to consume services from other servers. Isn't this just JWS BB version? It's been extremely successful.
Finally I think JWS is not popular is also about how Sun advertise it. BB never advertises how good their BB Java apps are, they believe clients won't even care what is it. BB advertises the benefits of using MDS to develop apps: Fast, Cost Saving, Business Return.
Just my, a bit long, 2 cents... :)