Traditionally, projects have followed a plan, define requirements, do, test, and deploy model. There is a lot of upfront planning and scoping the project. A plan is created to design and develop a product. This cycle takes a long time and has a period where the end user custom isn’t involved in the process. When the product finally is released, it’s often disliked by the customer.
In this case, replace product with anything, be it a report, a piece of software, or a physical product.
An iterative development process breaks the development cycle down into really small chunks. This allows the product to be released as soon as possible and kept in a beta state. In the first few iterations, a smaller group of customers, typically early adopters are involved in testing out the product. Feedback from the direct customers are used to address bugs and features of the product. Over time, the product is release to the mass audience and changes are made incrementally. Google’s Gmail is an excellent example of this approach. The beta tag was kept on the logo for years before finally releasing it as a final product.
The advantage of this process is that it hits the market sooner than later and feedback is given from the actual users. By releasing features slowly, it also helps with change management efforts.
The disadvantage is that the product is released with errors and bugs. There could be a negative perception on the quality of the product. However, this can be address by communicating the continual improvement process.
I believe that the traditional development process will be obsolete. Holding on to products until its perfect will no longer work. We’ve seen more acceptance of this model through mediums like blogs, twitter, and social media.
There will be some kind of balance between the linear approach, which tends to be recursive, and this iterative process (aka "agile development") – especially in industries/areas where health, safety and/or security are paramount.
I think there's always a balance between the two extremes. I would argue that even for industries where health, safety, and security are of concern, the product should be released to the working team in iteration so that it can be tested, revised, tested, and revised many times until it is ready for production.