Let me explain.
At the moment, the most prominent devices are iPhone, Android, and Blackberry. The iPhone is popular today, it's programmed in Objective-C, you can't really use anything but Apple's development tools for it, and you might have to start a war against Apple to have your app available in their market. Android is gaining a lot of ground, and will possibly be the most popular mobile OS on Earth; it's programmed in Java and while you are not forced to use their SDK, you don't really have a lot of choice. Blackberry is still around, but losing market share every day. Windows Mobile doesn't exist (yet) but tomorrow? Who knows.
So, when writing an application you have two options: the first one is to learn the software development tool of each one of the important mobile platforms available today (I'd say iPhone and Android), and keep track of whatever might be popular tomorrow (and develop for that platform too).
Each option has advantages and disadvantages -- like everything else in this field. This post clearly explains the pros and cons of each choice. In general, one of the major problems of online applications is that they are bound to only access whatever the browser has access to, and things like the mobile's gyroscope, the GPS, the accelerometer, are often (or always) out of reach.
I have to say, when people face the "online vs. native" dilemma, they often don't realise how much fragmentation there is in the mobile market. At the moment, in the Smartphone arena (which isn't the biggest one -- some people really want just phones) Android and iPhone are both "fashionable". Blackberries are also strong and fashionable in business circles. Thing is, with fashion you never know what's going to be cool next year: maybe Nokia's Meego? (Nokia is pretty big, and the software is based on the established Maemo and QT...). Maybe it will be Samsung's Bada (Samsung sells an intimidating number of smartphones and normal phones every year, have recently released the Android based Galaxy S which is possibly the sexiest and most powerful Android phone right now); maybe the next fashionable system will be Palm's WebOS... sure, it's unlikely, but in the mobile market you just never know: all it takes is one fantastic phone that sells really well.
(I feel I need to add something about Bada: I feel that it's quite possible that the market will be flooded by Bada devices sooner rather than later. Samsung is the biggest producer of cell phones, way ahead of its competition in terms of numbers. Right now there is a huge slice of the market owned by "normal" phones that cost between $200 and $400, and if Samsung decides that all those phones will effectively become "smart" and have Bada, then you have all the ingredients to have a big player in the mobile world).
In developing Apollo, we decided to use a mobile library; this is being tested as we speak, and it shouldn't be too much longer before it's available to all users. As the world of mobile development changes very quickly, we will probably release a 2.0 version of our Apollo mobile application based on a more advanced library in 2011 (exciting!).
We believed in the browser as a platform when we developed Apollo; we could have written yet another "page reload" project management web application -- instead, we took the road less travelled and created a very unique product. We are now taking a very similar approach for our mobile platform: it's again the road less travelled, but we will definitely take it again.