I was incorrect when I wrote that Moodle Mobile would have an Android-native client. Carlos Kiyan, a member of the Moodle Mobile team, clarified for me over Twitter that Moodle Mobile is a web app which has, for the moment, focused on WebKit-based browsers. (Both the iPhone and Android have default browsers that are WebKit-based.) Sorry for missing that; I should have looked at those videos more closely. Meanwhile, Kayvon Beykpour of the Blackboard Mobile Learn team has written a blog post that talks about their decision to go with native clients.
As you might expect, each team thinks their approach is better. The Moodle Mobile team thinks that the web-based approach is more portable and will allow them to reach more smart phones. They think that the main gap between web-based and native apps is working offline, and they have developed offline caching for their app using a third-party product. (HTML 5, which both Google and Apple are promoting, will support offline capabilities, but it is not clear when this will be practically available in mobile browsers.) The Blackboard Mobile Learn team claims that native apps will allow them to provide a better user experience. To be honest, I don't know much about mobile app development and have no intuitions about who is right. I have been impressed with the quality of Google's Buzz mobile web app, but I don't assume that the approach will work for everything that you'd want to do in an app. We'll have to see once the Blackboard Mobile Learn team has their app out whether they are doing interesting things that a web app can't do.
Regarding portability, there's a long term and a short term issue. In the short term, Blackboard will have a Blackberry app while Moodle won't. I suspect that the market share for Blackberries among college students is pretty low, even in the United States. On the other hand, it's not always about numbers. If, for example, you have an important executive education program offered by your business school, then Blackberry support will matter a lot. In the long term, I suspect that the web app approach will be more portable and allow the developers to keep up on more platforms with fewer resources. But we'll see. This is a very young market.
Update: It appears that Blackberry plans to have a WebKit-based browser in the near future.
Later Update: Carlos Kiyan pointed me in the direction of this article, which outlines the pros and cons of both native and web app approaches.